Gradient and Intercept

12 Key excerpts on "Gradient and Intercept"

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  Introductory Mathematics

  • J Daniels, M Kropman, J Daniels, M Kropman(Authors)
  • 2014(Publication Date)
  • Future Managers

    (Publisher)

  • The x -intercept • The y -intercept • The gradient (slope) Consider the gradient–intercept form of a straight-line graph y = mx + c . • The c -value is where the graph intersects (crosses) the y -axis, thus the y -intercept . • The m -value is the gradient of the graph and measures the slope or steepness of the line. x - and y -intercepts of a straight line x -intercept of a straight line y x on the y -axis: x = 0 on the x -axis: y = 0 The x -intercept is where the graph intersects the x -axis. This is the point on the graph where y = 0 . y -intercept of a straight line The x -coordinate on any point on the y -axis is 0 (zero). So, to find the y -intercept, let x = 0 in the equation and then solve for y . Gradient (slope) of a straight line The gradient ( m ) of a straight line is the measure of its slope relative to the x -axis . This is expressed as a ratio of its vertical change to its horizontal change. Both changes are measured on the scales of their respective axes. Explanation Diagram Examples In the diagram, the gradient is given by b a . The gradient is positive since y increases as x increases. y x b = change in y a = change in x 0 m = 2 1 = 2 y x 0 1 1 2 1 2 3 4 2 14243 14243 A negative gradient shows that y decreases when x increases. The gradient is measured as in the diagram above, but has a negative sign in front of the value. y x b a 0 m = – 3 2 y x 0 –1 –3 2 1 2 3 4 –2 14243 1442443 158 Module 5 • Algebraic graphs To read the gradient (slope) of a straight line, the equation must be in the gradient– intercept form y = mx + c . The sign of the gradient is the key to determine whether the gradient is positive or negative. If the gradient is positive or m > 0, it is a positive increasing function. (The line slopes up to the right, or have a positive slope.) If the gradient is negative or m < 0, it is a negative decreasing function.

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  Elementary Algebra

  • Alan Tussy, R. Gustafson(Authors)
  • 2012(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Of all of the ways in which a linear equation can be written, one form, called slope – intercept form, is probably the most useful. When an equation is written in this form, two important features of its graph are evident. Use Slope–Intercept Form to Identify the Slope and y -Intercept of a Line. To explore the relationship between a linear equation and its graph, let’s consider . To graph this equation, three values of x were selected ( 1, 0, and 1), the corresponding values of y were found, and the results were entered in the table. Then the ordered pairs were plotted and a straight line was drawn through them, as shown below. y 2 x 1 y 2 x 1 0 1 1 3 (1, 3) (0, 1) ( 1, 1) 1 1 ( x , y ) y x To find the slope of the line, we pick two points on the line, and , and draw a slope triangle and count grid squares: Slope rise run 2 1 2 (0, 1) ( 1, 1) x y y = 2 x + 1 2 1 (–1, –1) (0, 1) (1, 3) 1 –1 2 3 4 –2 –3 –4 –2 –3 –4 2 3 4 Copyright 201 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. From the equation and the graph, we can make two observations: ■ The graph crosses the -axis at 1. This is the same as the constant term in . ■ The slope of the line is 2. This is the same as the coefficient of in . This illustrates that the slope and -intercept of the graph of can be determined from the equation. The slope of the line is 2. The -intercept is . These observations suggest the following form of an equation of a line.

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  Mathematical Applications for the Management, Life, and Social Sciences

  • Ronald Harshbarger, James J. Reynolds(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Linear Function As in Section 1.1 with linear equations, a linear function is also called a first-degree func- tion, and the graph of such a function is a straight line. The point(s) where a graph intersects the x-axis are called the x-intercept points, and the x-coordinates of these points are the x-intercepts. Similarly, the points where a graph intersects the y-axis are the y-intercept points, and the y-coordinates of these points are the y-intercepts. Because any point on the x-axis has y-coordinate 0 and any point on the y-axis has x-coordinate 0, we find intercepts as follows. Intercepts (a) To find the y-intercept(s) of the graph of an equation, set x 5 0 in the equation and solve for y. Note: A function of x has at most one y-intercept. (b) To find the x-intercept(s), set y 5 0 and solve for x. Intercepts Because the graph of a linear function is a line, only two points are necessary to determine its graph. It is frequently possible to use intercepts to graph a linear function. Intercepts and Graphing Find the intercepts and graph the following. (a) 3x 1 y 5 9 (b) x 5 4y Solution (a) To find the y-intercept, we set x 5 0 and solve for y: 3(0) 1 y 5 9 gives y 5 9, so the y-intercept is 9. To find the x-intercept, we set y 5 0 and solve for x: 3x 1 0 5 9 gives x 5 3, so the x-intercept is 3. Using the intercepts gives the graph shown in Figure 1.11. EXAMPLE 1 SEC TION 1.3 OBJECTIVES • To find the intercepts of graphs • To graph linear functions • To find the slope of a line from its graph and from its equation • To find the rate of change of a linear function • To graph a line, given its slope and y-intercept or its slope and one point on the line • To write the equation of a line, given information about its graph Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

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  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  b. State the slope and vertical intercept of your line, including units. What do they tell you about the problem? SOLUTION a. We first find two data points, (x, C), from the information given. Step 1: Compute the slope of the line. m = C 2 − C 1 _ x 2 − x 1 = 15, 000 − 9,000 __ 125 − 50 = 80 Step 2: Apply the point-slope formula, using (50, 9,000) for (x 1, y 1 ). C − C 1 = m (x − x 1 ) C − 9,000 = 80(x − 50) = 9,000 + 80x − 4,000 = 80x + 5,000 b. The slope is 80 dollars per bike, and it tells us the cost of producing each bike. The vertical intercept is 5,000, and it tells us that the bicycle company’s fixed costs (before production begins) are $5,000. The following summary reminds us that all horizontal lines have equations of the form y = b, and slopes of 0. Since they cross the y-axis at b, the y-intercept is b; there is no x-intercept. Vertical lines have no slope, and equations of the form x = a. Each will have an x-intercept at a, and no y-intercept. Finally, equations of the form y = mx have graphs that pass through the origin. The slope is always m and both the x-intercept and the y-intercept are 0. EXAMPLE 7 FIGURE 6 C x 25 50 75 100 125 150 2 4 6 8 10 20 18 16 14 12 x C 50 9,000 125 15,000 286 Chapter 4 Slope, Rates of Change, and Linear Functions Piecewise Defined Functions A function may be defined by different formulas on different portions of the x-axis. Such a function is said to be defined piecewise. Graph the function defined by f (x) = { x + 1 3 if x ≤ 1 if x > 1 SOLUTION Think of the coordinate system as divided into two regions by the ver- tical line x = 1, as shown in Figure 8. In the left-hand region (x ≤ 1), we graph the line y = x + 1. Notice that the point (1, 2) is included in the graph. We indicate this with a solid dot at the point (1, 2). In the right-hand region (x > 1), we graph the horizontal line y = 3. The point (1, 3) is not part of the graph. We indicate this with an open circle at the point (1, 3).

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  Mathematics for Biological Scientists

  • Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
  • 2009(Publication Date)
  • Garland Science

    (Publisher)

  c, then:

  Slope   =  

  f

  ( 100 )

    −   f

  ( 0 )

  100   − 0

    =

  212   − 32

  100

    = 1.8 =  

  9 5

  .

  Interestingly, the value of the slope determined here is the same as the coefficient of the c term in EQ5.1 .

  More generally, we can try the same approach using some basic coordinate geometry and the generic formula for a straight line:

  y ( x )   =   A   +   B x .

  (EQ5.2)

  Choosing any two pairs of coordinates that lie on this line, (x1 , y1 ) and (x2 , y2 ), we can determine the slope (Figure 5.6 ). However, the coordinates of both of these points must satisfy EQ5.2 , which means that y1 = A + Bx1 and y2 = A + Bx2 . Using this information to calculate the slope, namely the rate of increase of y with respect to an increase in x, we get:

  Slope   =  

  y 2

    −  

  y 1

  x 2

    −  

  x 1

    =  

  (

  A   +   B

  x 2

  )

    −  

  (

  A   +   B

  x 1

  )

  x 2

    −  

  x 1

    =  

  B

  (

  x 2

    −  

  x 1

  )

  x 2

    −  

  x 1

    =   B .

  This result proves that when the equation of a straight line is written in the form y = A + Bx, the gradient is always represented by the value of the constant B.

  Figure 5.6The gradient (or slope) of a straight line is equal to (y2 – y1 )/(x2 – x1 ), the increase in y values divided by the increase in x values.

  Figure 5.7The position of an apple at 0, 20, 40, and 60 ms after it begins its descent toward the head of Isaac Newton.

  Figure 5.8A plot of y against x for y(x) = 0.05x2 . The gradient of the straight line between points at x = 40 and x

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  Explorations in College Algebra

  • Linda Almgren Kime, Judith Clark, Beverly K. Michael(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Why? Many models exclude negative numbers (like the prior examples on infant weight or computer depreciation), so the smallest value in the domain is often 0. When x = 0, then y b m = + ⋅ 0 y b = The point (0, b) satisfies the equation and is the vertical intercept. Since the coordinate b tells us where the line crosses the y-axis, we often just refer to b as the vertical or y-intercept (see Figure 2.19). A Linear Function A function y = ƒ(x) is called linear if it can be represented by an equation of the form y b mx = + Its graph is a straight line where m is the slope, the rate of change of y with respect to x. So if (x 1 , y 1 ) and (x 2 , y 2 ) are any two distinct points on the line, then the y y x x y x m slope average rate of change 2 1 2 1 = − − = D D = = b is the vertical or y-intercept and is the value of y when x = 0. If the function is a math- ematical model, b typically represents an initial or starting value. The equation y = b + mx could, of course, be written in the equivalent form y = mx + b. This may be the traditional format that you first learned. EXAMPLE 4 Identifying the Vertical Intercept and Slope For each of the following equations, identify the value of b and the value of m. a. y = −4 + 3.25x c. g(x) = −4x + 3.25 b. f(x) = 3.25x − 4 d. y = 3.25 − 4x b (vertical intercept) m (slope) = Dy Dy D x D x y x FIGURE 2.19 Graph of y = b + mx. 88 CHAPTER 2 Rates of Change and Linear Functions Solution a. b = −4 and m = 3.25 b. b = −4 and m = 3.25 c. b = 3.25 and m = −4 d. b = 3.25 and m = −4 EXAMPLE 5 Comparing Legal Costs In the following equations, L(h) represents the legal fees (in dollars) charged by four different law firms and h represents the number of hours of legal advice. L h h L h h L h h L h h ( ) 500 200 ( ) 800 350 ( ) 1000 150 ( ) 500 1 3 2 4 = + = + = + = a. Which initial fee is the highest? b. Which rate per hour is the highest? c. If you need 5 hours of legal advice, which legal fee will be the highest? Solution a.

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  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 1 L i n e a r F u n c t i o n s 44 d. We locate the output of y 5 23.5 on the vertical axis and trace over to the line. 0 5 4 3 2 1 ]2 ]3 ]4 ]5 ]5 ]4 ]3 ]2 ]1 1 2 3 4 5 y x ]1 The line has an output of y 5 23.5 when the input is x 5 23. e. We locate the input value of x 5 22 on the horizontal axis and trace down to the line. 0 5 4 3 2 1 ]2 ]3 ]4 ]5 ]5 ]4 ]3 ]2 ]1 1 2 3 4 5 y x ]1 The line has an output of y 5 23 when the input is x 5 22. PRACTICE PROBLEM FOR EXAMPLE 2 Use the graph to make the following estimates. 0 25 20 15 10 5 ]10 ]15 ]20 ]25 ]20 ]10 10 20 y x ]5 a. Estimate the y-intercept. b. Estimate the x-intercept. c. Estimate the value of x that results in y 5 210. d. Estimate the value of y when x 5 215. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. S E C T I O N 1 . 3 I n t e r c e p t s a n d G r a p h i n g 45 Explaining the meaning of the intercepts of a line is an important part of understanding what role they play in an application. The intercepts of a line are points on the graph, so they represent combinations of input and output values that satisfy the equation. Both parts of each intercept have a meaning and should be explained in terms of the problem situation.

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  Functions Modeling Change

  A Preparation for Calculus

  • Eric Connally, Deborah Hughes-Hallett, Andrew M. Gleason(Authors)
  • 2019(Publication Date)
  • Wiley

    (Publisher)

  Formulas for the Equation of a Line Summarizing, the following equations are all used to represent lines: • The slope-intercept form is  =  +  where  is the slope and  is the -intercept. • The point-slope form is  −  0 = ( −  0 ) where  is the slope and ( 0 ,  0 ) is a point on the line. • The standard form is  +  =  where , , and  are constants. Equations of Horizontal and Vertical Lines The slope  of a line  =  +  gives the rate of change of  with respect to . What about a line with slope  = 0? If the rate of change of a quantity is zero, then the quantity does not change. Thus, if the slope of a line is zero, the value of  must be constant. Such a line is horizontal. Example 5 Explain why the equation  = 4 represents a horizontal line and the equation  = 4 represents a vertical line. Solution The equation  = 4 represents a linear function with slope  = 0. To see this, notice that this equation can be rewritten as  = 4 + 0 ⋅ . Thus, the value of  is 4 no matter what the value of  is. See Figure 1.35. Similarly, the equation  = 4 means that  is 4 no matter what the value of  is. Every point on the line in Figure 1.36 has  equal to 4, so this line is the graph of  = 4. −2 −1 1 2 1 2 3 5 6 7 8 (−2, 4) (−1, 4) (1, 4) (2, 4)   ✠ Value of  is 4 no matter what the value of  is ❄ ❄ Figure 1.35: The horizontal line  = 4 has slope 0 1 2 3 4 5 6 7 8 1 2 3   (4, 3) ✛ Value of  is 4 no matter what the value of  is (4, 1) ✙ ✠ (4, −1) Figure 1.36: The vertical line  = 4 has an undefined slope What is the slope of a vertical line? Figure 1.36 shows three points, (4, −1), (4, 1), and (4, 3), on a vertical line. Calculating the slope gives  = Δ Δ = 3 − 1 4 − 4 = 2 0 . 34 Chapter 1 LINEAR FUNCTIONS AND CHANGE The slope is undefined because the denominator, Δ, is 0. The slope of every vertical line is undefined for the same reason. A vertical line is not the graph of a function, since it fails the vertical line test.

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  Painless Pre-Algebra

  • Barron's Educational Series, Amy Stahl(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  y-intercept. What do they mean in the context of the problem? Graph the line.

  (Answers are on pages 121 –123 .)

  A Closer Look at y = mx + b

  Now that you have reviewed how to graph a line using slope-­intercept form, let’s look at the equation a little bit more. Compare the equations in the two-column table below. What do you notice?

  Both equations are very similar. y = mx + b has a slope and a y-intercept. y = mx has a slope and a y-intercept of ZERO; hence, you never see the b.

  It is important to remember that both of these equations graph lines (linear functions). But if a line graph passes through (0, 0), it has a special name. It is called a direct variation. And this means the slope is also proportional.

  Let’s look at these two big ideas. Compare these two tables. Let’s find the slope of each table. Both equations have the same slope. Now look at each graph. You can also look at the tables to see if the points will reach (0, 0). If you follow the pattern for slope for each table, notice the tables also prove that Table A will pass through (0, 0), but Table B will not.

  Since Table A is a direct variation, it is also proportional. To determine if two quantities are proportional, compare the ratio of for several pairs of points.

  Look at the tables again.

  MATH TALK!

  If a line is a direct variation, it will pass through (0, 0), and it will be proportional (each pair will reduce to the same ratio).

  KEY CONCEPT: You can use tables, words, equations, or graphs to compare proportional relationships.

  TABLE:

  WORDS: Constant of proportionality =

  EQUATION: y-intercept = 0

  GRAPH: will pass through (0, 0)

  CAUTION—Major Mistake Territory!

  When checking for proportionality, make sure you always do for each pair of points!

  BRAIN TICKLERSSet # 23

  Determine whether each linear function is a direct variation. If so, state the constant of proportionality.

  1.

  2.

  3.

  4.

  5.

  6.

  In each case, y varies directly as x

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  Precalculus

  A Self-Teaching Guide

  • Steve Slavin, Ginny Crisonino(Authors)
  • 2001(Publication Date)
  • Trade Paper Press

    (Publisher)

  y.

  The y-intercept is (0, −4).

  We’ll find the x-intercepts by letting y = 0 and solving for x.

  This is a quadratic equation (second degree—highest exponent is 2) that doesn’t factor, so we’ll use the quadratic formula

  The x-intercepts are

  Example 4:

  Find the intercepts of .

  Solution:

  Let’s find the y-intercept by substituting 0 for x and solving for y.

  The y-intercept is the point (0, −36).

  We’ll find the x-intercept by substituting 0 for y and solving for x. We know that the only way a fraction can equal 0 is if the numerator equals 0, so we’ll set the numerator equal to 0 and solve for y.

  Factor using the difference of two squares.

  Set each factor equal to 0 and solve for x.

  The x-intercepts are (−6, 0) and (6, 0).

  SELF-TEST 1:

  Find the intercepts for each of the following functions:

  1.

  2.

  3.

  4.

  5.

  6.

  7.

  8.

  ANSWERS:

  1.

  2.

  3.

  4.

  5.

  6.

  7.

  8. x-intercept: none (this graph does not cross the x-axis)

  This is a fraction, so the numerator has to equal 0 for the fraction to equal 0, but because the numerator doesn’t have a variable in it the numerator will always be 10, never 0. Therefore there isn’t any x-intercept.

  2 Slope of a Straight Line

  The slope of a line is a measure of how steeply a line rises or falls. It’s actually a ratio of the vertical change to the horizontal change between two points on the graph. The vertical change is the change in the y coordinates between two points on the graph, and the horizontal change is the change in the x coordinates between two points on the graph. We’ll refer to the two points as (x2 , y2 ) and (x1 , y1 ). In mathematical notation, the Greek letter Δ (delta) means “change.” The symbol for slope is m.

  When we use our slope formula it doesn’t make any difference which one we call the first point and which one we call the second point; we’ll still get the same slope when we use the slope formula

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  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  190 CHAPTER 3 Linear Equations in Two Variables and Functions The slope of this line is 2. The line we are interested in is perpendicular to the line with slope 2 and must, therefore, have a slope of − 1 _ 2 . Using ( x 1 , y 1 ) = ( − 1, 4) and m = − 1 _ 2 , we have y − y 1 = m ( x − x 1 ) y − 4 = − 1 __ 2 ( x + 1) Because we want our answer in standard form, we multiply each side by 2. 2 y − 8 = − 1( x + 1) 2 y − 8 = − x − 1 x + 2 y − 8 = − 1 x + 2 y = 7 The last equation is in standard form. As a final note, the following summary reminds us that all horizontal lines have equations of the form y = b , and slopes of 0. Since they cross the y -axis at b , the y -intercept is b ; there is no x -intercept. Vertical lines have no slope, and equations of the form x = a . Each will have an x -intercept at a , and no y -intercept. Finally, equations of the form y = mx have graphs that pass through the origin. The slope is always m and both the x -intercept and the y -intercept are 0. Find the equation of the line passing through the points ( − 2, − 3) and ( − 2, 5). SOLUTION First, we find the slope of the line. m = 5 − (− 3) _________ − 2 − (− 2) = 8 __ 0 , which is undefined Special Equations: Their Graphs, Slopes, and Intercepts For the equations below, m, a, and b are real numbers. Through the Origin Vertical Line Horizontal Line Equation: y = mx Equation: x = a Equation: y = b Slope = m Undefined slope Slope = 0 x -intercept = 0 x -intercept = a No x -intercept y -intercept = 0 No y -intercept y -intercept = b FIGURE 5a FIGURE 5b FIGURE 5c x y Through the Origin y H11005 mx x y Vertical Line x H11005 a a x y Horizontal Line y H11005 b b FACTS FROM GEOMETRY EXAMPLE 8 3.3 The Equation of a Line 191 Because there is no slope, we cannot use slope-intercept form or point-slope form. A line having no slope must be a vertical line. The graph of the line is shown in Figure 6.

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  Calculus of a Single Variable: Early Transcendental Functions, International Metric Edition

  • Ron Larson, Bruce Edwards(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  16 Chapter 1 Preparation for Calculus 1.2 Exercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. CONCEPT CHECK 1. Slope-Intercept Form In the form y = mx + b, what does m represent? What does b represent? 2. Perpendicular Lines Is it possible for two lines with positive slopes to be perpendicular? Why or why not? Estimating Slope In Exercises 3–6, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com. 3. x 1 2 3 4 5 6 7 7 6 5 4 3 2 1 y 4. x 1 2 3 4 5 6 7 7 6 5 3 2 1 y 5. x 1 2 3 4 5 6 6 5 4 3 2 1 y 6. x 1 2 3 5 6 7 24 28 20 16 12 8 4 y Finding the Slope of a Line In Exercises 7–12, plot the pair of points and find the slope of the line passing through them. 7. (3, -4), (5, 2) 8. (0, 0), (-2, 3) 9. (4, 6), (4, 1) 10. (3, -5), (5, -5) 11. (- 1 2 , 2 3 ), (- 3 4 , 1 6 ) 12. ( 7 8 , 3 4 ), ( 5 4 , - 1 4 ) Sketching Lines In Exercises 13 and 14, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes 13. (3, 4) (a) 1 (b) -2 (c) - 3 2 (d) Undefined 14. (-2, 5) (a) 3 (b) -3 (c) 1 3 (d) 0 Finding Points on a Line In Exercises 15–18, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) Point Slope Point Slope 15. (6, 2) m = 0 16. (-4, 3) m is undefined. 17. (1, 7) m = -3 18. (-2, -2) m = 2 Finding an Equation of a Line In Exercises 19–24, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope 19. (0, 3) m = 3 4 20. (-5, -2) m = 6 5 21. (1, 2) m is undefined. 22. (0, 4) m = 0 23. (3, -2) m = 3 24. (-2, 4) m = - 3 5 25. Road Grade You are driving on a road that has a 6% uphill grade. This means that the slope of the road is 6 100 . Approximate the amount of vertical change in your position when you drive 200 meters.

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