Solving and Graphing Quadratic Equations

11 Key excerpts on "Solving and Graphing Quadratic Equations"

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  Intermediate Algebra

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

    (Publisher)

  In this section, we will explore further these types of functions and their graphs. Up to now in this chapter, we have been studying quadratic equations. If we take a quadratic equation in standard form and use it instead as the formula for a function, the result is called a quadratic function . Here is a formal definition: Any function that can be written in the form f ( x ) = a x 2 + bx + c where a , b , and c are constants with a ≠ 0, is called a quadratic function . We refer to this form as standard form . quadratic function DEFINITION The Basic Quadratic Function The simplest of all quadratic functions results when we let a = 1 and b = c = 0. We will refer to f ( x ) = x 2 as the basic quadratic function . Table 1 gives some ordered pairs for this function, and the corresponding graph is shown in Figure 1. This graph is an example of a parabola . From the graph we can see that the domain of this function is the set of all real numbers and the range is { y | y ≥ 0}. The point (0, 0) where the parabola changes direction (the lowest point on the parabola) is called the vertex . FIGURE 1 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 Vertex y 5 x 2 2 3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 1 2 3 5 6 7 8 9 x y Vertex f ( x ) = x 2 Axis of Symmetry Axis of Symmetry 4 x f ( x ) = x 2 − 3 9 − 2 4 − 1 1 0 0 1 1 2 4 3 9 TABLE 1 608 CHAPTER 8 Quadratic Equations and Functions We also observe that the graph is symmetric about the y -axis, meaning the right half of the graph is a mirror image of the left half. For this reason, the vertical line x = 0 passing through the vertex is called the axis of symmetry . All quadratic functions have a graph that is a parabola and a domain that is the set of all real numbers. However, the shape, direction, and position of the parabola can vary as we will see in the following segment. Transformations Let's consider quadratic functions of the form f ( x ) = a x 2 .

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  Technical Mathematics with Calculus

  • Paul A. Calter, Michael A. Calter(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  125  15.5t  16.1t 2 s  v 0 t  1 2 at 2 a  32.2 ft/s 2 v 0  15.5 ft/s, s  125 ft, FIGURE 12–1 12–1 Solving a Quadratic Equation Graphically and by Calculator Recall that a polynomial equation is one in which all powers of x are positive inte- gers. A quadratic equation is a polynomial equation of second degree. That is, the highest power of x in the equation is 2. It is common practice to refer to a quadratic equation simply as a quadratic. ◆◆◆ Example 1: The following equations are quadratic equations: (a) (b) (c) (d) (e) ◆◆◆ A quadratic function is one whose highest-degree term is of second degree. ◆◆◆ Example 2: The following functions are quadratic functions: (a) (b) (c) (d) ◆◆◆ Some quadratic equations have a term missing. A quadratic that has no x term is called a pure quadratic; one that has no constant term is called an incomplete quadratic. ◆◆◆ Example 3: (a) is a pure quadratic. (b) is an incomplete quadratic. ◆◆◆ Solving a Quadratic Graphically We will show several ways to solve a quadratic; first graphically and by calculator, and, in the next section, by formula. ■ Exploration: In our chapter on graphing we plotted the quadratic function getting a curve that we called a parabola. Try this. Either graph this function again or look back at our earlier graph. Does the curve intercept the x axis, and if so, how many times? Can you imagine a parabola that has more x-intercepts? Zoom out far enough to convince yourself that the curve will not turn and re-cross the x axis at some other place. Can you imagine a parabola that has fewer x-intercepts? Using , or zero, find the value of x at an intercept and substitute it into the given function. Then state the significance of an intercept. ■ Your exploration may have shown you that a quadratic function in x can have 0, 1, or 2 x-intercepts, but no more than two. Also, the value of x at an intercept, when substituted back into the function, makes that function equal to zero.

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  Algebra

  A Combined Course 2E

  • Charles P. McKeague(Author)
  • 2018(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter 12 Quadratic Equations and Functions 990 Equations Quadratic in Form [12.4] There are a variety of equations whose form is quadratic. We solve most of them by making a substitution so the equation becomes quadratic, and then solving the equation by factoring or the quadratic formula. For example, The equation is quadratic in (2x − 3) 2 + 5(2x − 3) − 6 = 0 2x − 3 4x 4 − 7x 2 − 2 = 0 x 2 2x − 7 √ — x + 3 = 0 √ — x Graphing Quadratic Functions [12.5] The graph of any function of the form f (x) = ax 2 + bx + c a ≠ 0 is a parabola. The graph is concave up if a > 0 and concave down if a < 0. The highest or lowest point on the graph is called the vertex and always has an x-coordinate of x = −b ___ 2a . Quadratic Inequalities [12.6] We solve quadratic inequalities by manipulating the inequality to get 0 on the right side and then factoring the left side. We then make a diagram that indicates where the factors are positive and where they are negative. From this sign diagram and the original inequality we graph the appropriate solution set. Rational Inequalities [12.6] We solve rational inequalities by simplifying the rational expressions on the left side and obtaining 0 on the right side. We then make a diagram that indicates where the factors are positive and where they are negative. From this sign diagram and the original inequality we graph the appropriate solution set, making sure to avoid zero denominators. 5. The equation x 4 − x 2 − 12 = 0 is quadratic in x 2 . Letting y = x 2 we have y 2 − y − 12 = 0 ( y − 4)( y + 3) = 0 y = 4 or y = −3 Resubstituting x 2 for y, we have x 2 = 4 or x 2 = −3 x = ±2 x = ±i √ — 3 6. The graph of y = x 2 − 4 will be a parabola. It will cross the x-axis at 2 and −2, and the vertex will be (0, −4). 7. Solve x 2 − 2x − 8 > 0. We factor and draw the sign diagram. (x − 4)(x + 2) > 0 The solution set is (−∞, −2) ∪ (4, ∞). x - 4 -2 4 x + 2 - - - - - - - - - - - - + + + + + + + + + + + + 8.

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  Elementary Algebra 2e

  • Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
  • 2020(Publication Date)
  • Openstax

    (Publisher)

  Chapter 10 Quadratic Equations 1231 Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? 1232 Chapter 10 Quadratic Equations This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 axis of symmetry completing the square consecutive even integers consecutive odd integers discriminant parabola quadratic equation quadratic equation in two variables Square Root Property vertex x-intercepts of a parabola y-intercept of a parabola CHAPTER 10 REVIEW KEY TERMS The axis of symmetry is the vertical line passing through the middle of the parabola y = ax 2 + bx + c. Completing the square is a method used to solve quadratic equations. Consecutive even integers are even integers that follow right after one another. If an even integer is represented by n , the next consecutive even integer is n + 2 , and the next after that is n + 4 . Consecutive odd integers are odd integers that follow right after one another. If an odd integer is represented by n , the next consecutive odd integer is n + 2 , and the next after that is n + 4 . In the Quadratic Formula, x = −b ± b 2 − 4ac 2a the quantity b 2 − 4ac is called the discriminant. The graph of a quadratic equation in two variables is a parabola. A quadratic equation is an equation of the form ax 2 + bx + c = 0 , where a ≠ 0. A quadratic equation in two variables, where a, b, and c are real numbers and a ≠ 0 is an equation of the form y = ax 2 + bx + c. The Square Root Property states that, if x 2 = k and k ≥ 0 , then x = k or x = − k. The point on the parabola that is on the axis of symmetry is called the vertex of the parabola; it is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. The x-intercepts are the points on the parabola where y = 0.

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  Introductory Algebra

  Concepts with Applications

  • Charles P. McKeague(Author)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  One method of graphing the solution set is to find a number of ordered pairs that satisfy the equation and to graph them. We can obtain some ordered pairs that are solutions to y = x 2 − 3 by use of a table as follows: Graphing these solutions and then connecting them with a smooth curve, we have the graph of y = x 2 − 3. (See Figure 1.) x y = x 2 − 3 y Solutions −3 y = (−3) 2 − 3 = 9 − 3 = 6 6 (−3, 6) −2 y = (−2) 2 − 3 = 4 − 3 = 1 1 (−2, 1) −1 y = (−1) 2 − 3 = 1 − 3 = −2 −2 (−1,−2) 0 y = 0 2 − 3 = 0 − 3 = −3 −3 (0,−3) 1 y = 1 2 − 3 = 1 − 3 = −2 −2 (1,−2) 2 y = 2 2 − 3 = 4 − 3 = 1 1 (2, 1) 3 y = 3 2 − 3 = 9 − 3 = 6 6 (3, 6) FIGURE 1 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 ( 2 3, 6) ( 2 2, 1) ( 2 1, 2 2) (3, 6) (1, 2 2) (0, 2 3) Vertex y 5 x 2 2 3 -5-4-3-2-1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 x y (-3, 6) (-2, 1) (-1, -2) (3, 6) (2, 1) (1, -2) (0, -3) Vertex y = x 2 - 3 Image © Katherine Heistand Shields, 2010 Chapter 9 Quadratic Equations and Functions 660 This graph is an example of a parabola. All equations of the form y = ax 2 + bx + c, a ≠ 0, have vertical parabolas for graphs. Although it is always possible to graph parabolas by making a table of values of x and y that satisfy the equation, there are other methods that are faster and, in some cases, more accurate. The important points associated with the graph of a parabola are the highest (or lowest) point on the graph and the x-intercepts. The y-intercept can also be useful. A Intercepts for Vertical Parabolas The graph of the function y = ax 2 + bx + c crosses the y-axis at y = c, because when x =0, y = a(0) 2 + b(0) + c = c. Because the graph crosses the x-axis when y = 0, the x-intercepts are those values of x that are solutions to the quadratic equation 0 = ax 2 + bx + c. The Vertex of a Parabola The highest or lowest point on a parabola is called the vertex.

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  Elementary and Intermediate Algebra

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

    (Publisher)

  We refer to this form as standard form . quadratic function DEFINITION The Basic Quadratic Function The simplest of all quadratic functions results when we let a = 1 and b = c = 0. We will refer to f ( x ) = x 2 as the basic quadratic function . Table 1 gives some ordered pairs for this function, and the corresponding graph is shown in Figure 1. This graph is an example of a parabola . FIGURE 1 –5 –4 –3 –2 –1 1 2 3 4 5 –5 –4 –3 –2 –1 1 2 3 4 5 Vertex y 5 x 2 2 3 -5 -4 -3 -2 -1 1 2 3 4 5 -1 1 2 3 5 6 7 8 9 x y Vertex f ( x ) = x 2 Axis of Symmetry Axis of Symmetry 4 x f ( x ) = x 2 − 3 9 − 2 4 − 1 1 0 0 1 1 2 4 3 9 TABLE 1 820 Chapter 11 Quadratic Equations and Functions From the graph we can see that the domain of this function is the set of all real numbers and the range is { y | y ≥ 0}. The point (0, 0) where the parabola changes direction (the lowest point on the parabola) is called the vertex . We also observe that the graph is symmetric about the y -axis, meaning the right half of the graph is a mirror image of the left half. For this reason, the vertical line x = 0 passing through the vertex is called the axis of symmetry . All quadratic functions have a graph that is a parabola and a domain that is the set of all real numbers. However, the shape, direction, and position of the parabola can vary as we will see in the following segment. Transformations Let's consider quadratic functions of the form f ( x ) = a x 2 . In this case, all we are doing is taking each y -value (output) from the basic quadratic function y = x 2 and multiplying it by a factor of a . As a result, we can change the shape and direction of the basic parabola. We illustrate how this is done in the next two examples. Graph: f ( x ) = 2 x 2 . SOLUTION Because a = 2, we take each y -coordinate from the basic parabola y = x 2 and double it. Table 2 shows how this is done for several values of x .

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  Technical Mathematics with Calculus

  • Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  ◆◆◆ Example 21: The following functions are quadratic functions: (a) f (x) = 5x 2 - 3x + 2 (c) f (x) = x(x + 7) (b) f (x) = 9 - 3x 2 (d) f (x) = x - 4 - 3x 2 The Parabola When we plot a quadratic function, we get the well-known and extremely useful type of curve called the parabola. ◆◆◆ Example 22: Plot the quadratic function y = x 2 + x - 3 for x = -3 to x = 3. Solution: We can graph the function using a graphics calculator or graphing software on a computer, or by computing a table of point pairs as was shown in Sec. 5–2. x -3 -2 -1 0 1 2 y 3 -1 -3 -3 -1 3 We plot these points and connect them with a smooth curve, a parabola, as shown in Fig. 14-3. Note that the curve in this example is symmetrical about the line x 1 2 = - . This line is called the axis of symmetry. The point where the parabola crosses the axis of symmetry is called the vertex. Recall that in Chapter 5 we said that a low point in a curve is called a mini- mum point, and that a high point is called a maximum point. Thus the vertex of this parabola is also a minimum point. ◆◆◆ The parabola is one of the four conic sections (the curves obtained when a cone is intersected by a plane at various angles), and it has many interesting properties and applications. We take only a very brief look at the parabola here, with the main treatment saved for our study of the conic sections. ◆◆◆ 3.0 cm 3.0 cm FIGURE 14-2 x y Axis of symmetry x = - 1 2 3 2 2 1 1 -1 -1 -2 -2 -3 -3 0 Zero Zero Vertex y = x 2 + x - 3 FIGURE 14-3 315 Section 14–5 ◆ Graphing the Quadratic Function Solving Quadratics Graphically In Sec. 4–5 we saw that a graphical solution to the equation f (x) = 0 could be obtained by plotting the function y = f (x) and locating the points where the curve crosses the x axis. Those points are called the zeros of the function. The x coordinates of those points are then the solutions, or roots, of the equation. Notice that in Fig. 14-3 there are two zeros, corresponding to the two roots of the equation.

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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 4 Q u a d r a t i c F u n c t i o n s 418 We solved the quadratic equation for any values for a ? 0, b, and c. Thus, the formula works for any real values of a ? 0, b, and c. The final result is called the quadratic formula. x 5 2b 6 !b 2 2 4ac 2a a ? 0 This formula can be used to solve any quadratic equation in standard form as long as the equation is set equal to zero. Steps to Solving Quadratic Equations Using the Quadratic Formula 1. Set the quadratic equation equal to zero. 2. Rewrite the quadratic expression in standard form. 3. Substitute the values of a, b, and c into the quadratic formula. x 5 2b 6 !b 2 2 4ac 2a 4. Simplify the quadratic formula. Give exact and/or approximate answers per problem instructions. 5. Check answers in the original equation. Quadratic Equations Square Root Property Use when there is a squared term but no first-degree term. Isolate the squared term and use a plus or minus symbol to indicate both answers. x 2 5 25 x 5 6 !25 x 5 65 Completing the Square Use if the vertex form is required. Finish solving with the square root property. x 2 1 6x 1 4 5 0 x 2 1 6x 5 24 x 2 1 6x 1 9 5 24 1 9 x 2 1 6x 1 9 5 5 1 x 1 32 2 5 5 Factoring Use when the quadratic equation has small coefficients that factor easily. Set the quadratic equal to zero, factor, and then use the zero factor property to write two or more simpler equations. x 2 1 7x 1 10 5 0 1 x 1 521 x 1 22 5 0 x 1 5 5 0 or x 1 2 5 0 Quadratic Formula Use when the quadratic equation has fractions, decimals, or large numbers.

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  Intermediate Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Doing this will ensure that you do your best in all your classes. Chapter 7 Summary 641 The Square Root Property for Equations [7.1] If a 2 = b, where b is a real number, then a = √ — b or a = − √ — b which can be written as a = ± √ — b . To Solve a Quadratic Equation by Completing the Square [7.1] Step 1: Write the equation in the form ax 2 + bx = c. Step 2: If a ≠ 1, divide through by the constant a so the coefficient of x 2 is 1. Step 3: Complete the square on the left side by adding the square of 1 _ 2 and the coefficient of x to both sides. Step 4: Write the left side of the equation as the square of a binomial. Simplify the right side if possible. Step 5: Apply the square root property for equations, and solve as usual. The Quadratic Theorem [7.2] For any quadratic equation in the form ax 2 + bx + c = 0, a ≠ 0, the two solutions are x = −b ± √ — b 2 − 4ac ______________ 2a This last equation is known as the quadratic formula. The Discriminant [7.3] The expression b 2 − 4ac that appears under the radical sign in the quadratic formula is known as the discriminant. We can classify the solutions to ax 2 + bx + c = 0: The solutions are When the discriminant is Two complex numbers containing i Negative One rational number Zero Two rational numbers A positive perfect square Two irrational numbers A positive number, but not a perfect square EXAMPLES 1. If (x − 3) 2 = 25 then x − 3 = ±5 x = 3 ± 5 x = 8 or x = −2 2. Solve x 2 − 6x − 6 = 0 x 2 − 6x = 6 x 2 − 6x + 9 = 6 + 9 (x − 3) 2 = 15 x − 3 = ± √ — 15 x = 3 ± √ — 15 3. If 2x 2 + 3x − 4 = 0, then x = −3 ± √ — 9 − 4(2)(−4) _________________ 2(2) = −3 ± √ — 41 _________ 4 4. The discriminant for x 2 + 6x + 9 = 0 is D = 36 − 4(1)(9) = 0, which means the equation has one rational solution. 642 CHAPTER 7 Quadratic Functions Equations Quadratic in Form [7.4] There are a variety of equations whose form is quadratic.

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  Beginning 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. ● ● A quadratic equation in two variables has the form y 5 ax 2 1 bx 1 c , where a ? 0 . ● ● A parabola is the shape of the graph of a quadratic equation in two variables. ● ● For a quadratic equation in the form y 5 ax 2 1 bx 1 c , if a . 0 (positive), the parabola opens upward, and if a , 0 (negative), the parabola opens downward. ● ● The vertex of a parabola is the lowest point on the graph for a parabola that opens upward, and it is the highest point on the graph for a parabola that opens downward. The vertex of a parabola that opens upward is called the minimum point . The vertex of a parabola that opens downward is called the maximum point . ● ● The x -coordinate of the vertex of the quadratic equation in two variables, y 5 ax 2 1 bx 1 c , is given by x 5 2 b 2 a . The y -coordinate is found by substituting the x -coordinate into the equation and simplifying. ● ● To interpret the meaning of the vertex in an application, be sure to explain what both coordinates of the vertex mean. Indicate if the vertex is a maximum or minimum point. ● ● The vertical line that passes through the vertex of a parbola is called the axis of symmetry . The axis of symmetry formula is x 5 2 b 2 a . ● ● To graph a quadratic equation in two variables , use these steps: 1. Find the equation of the axis of symmetry. 2. Find the coordinates of the vertex. 3. Graph a total of seven points. Find the values of additional symmetric points as necessary. There should be at least three points to the left and right of the vertex.

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  Elementary Algebra

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

    (Publisher)

  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. Before moving on to a new mathematics course, it’s worthwhile to take some time to reflect on your effort and performance in this course. Study Skills Workshop Preparing for Your Next Math Course As this course draws to a close, here are some questions to ask yourself. 1. How was my attendance? 2. Was I organized? Did I have the right materials? 3. Did I follow a regular schedule? 4. Did I pay attention in class and take good notes? 5. Did I spend the appropriate amount of time on homework? 6. How did I prepare for tests? Did I have a test-taking strategy? 7. Was I part of a study group? If not, why not? If so, was it worthwhile? 8. Did I ever seek extra help from a tutor or from my instructor? 9. In what topics was I the strongest? In what topics was I the weakest? 10. If I had it to do over, would I do anything differently? Now Try This SECTION 9.1 Solving Quadratic Equations: The Square Root Property Use the square root property to solve equations of the form . Use the square root property to solve equations of the form . Solve problems modeled by quadratic equations. ( ax b ) 2 c x 2 c OBJECTIVES 1. Factor: 2. How many square roots does 9 have? What are they? 3. Simplify: 4. Rationalize the denominator: 5. Simplify: 6. Approximate to the nearest hundredth: 2 51 B 23 16 B 5 7 2 75 x 2 64 ARE YOU READY? The following problems review some basic skills that are needed when solving quadratic equations. Recall that a quadratic equation can be written in the form , where , , and represent real numbers and . Some examples of quadratic equations are , , and We have solved quadratic equations like these using factoring in combination with the zero-factor property. To review this method, let’s solve . Factor the difference of two squares.

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