Section 6: Quadratic Equations and Functions Part 2

Section 6: Quadratic Equations and Functions Part 2 Topic 1: Observations from a Graph of a Quadratic Function... 147 Topic 2: Nature of the Solutions of Quadratic Equations and Functions... 150 Topic 3: Graphing Quadratic Functions Using a Table... 153 Topic 4: Graphing Quadratic Functions Using the Vertex and Intercepts... 156 Topic 5: Graphing Quadratic Functions Using Vertex Form - Part 1... 159 Topic 6: Graphing Quadratic Functions Using Vertex Form - Part 2... 161 Topic 7: Transformations of the Dependent Variable of Quadratic Functions... 164 Topic 8: Transformations of the Independent Variable of Quadratic Functions... 166 Topic 9: Finding Solution Sets to Systems of Equations Using Tables of Values and Successive Approximations... 169 Visit AlgebraNation.com or search "Algebra Nation" in your phone or tablet's app store to watch the videos that go along with this workbook! 145

The following Mathematics Florida Standards will be covered in this section: A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A-CED.1.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.2.4b - Solve quadratic equations in one variable. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as aa ± bbbb for real numbers aa and bb. A-REI.4.11 - Explain why the xx-coordinates of the points where the graphs of the equations yy = ff(xx) and yy = gg(xx) intersect are the solutions of the equation ff(xx) = gg(xx); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ff(xx) and/or gg(xx) are linear, rational, absolute value, and exponential functions. A-SSE.1.1b - Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret PP(1 + rr) 1 as the product of PP and a factor not depending on PP. F-BF.2.3 - Identify the effect on the graph of replacing ff(xx) by ff(xx) + kk, kkkk(xx), ff(kkkk), and ff(xx + kk) for specific values of kk (both positive and negative); find the value of kk given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F-IF.2.4 - For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.3.7a - Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology in more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. F-IF.3.8a - Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F-IF.3.9 - Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 146

Section 6: Quadratic Equations and Functions Part 2 Section 6 Topic 1 Observations from a Graph of a Quadratic Function Let s Practice! 1. The graph shows the height of a rocket from the time it was launched from the ground. Use the graph to answer the questions below. Let s review some things we learned earlier about the information we can gather from the graph of a quadratic. Vertex: Axis of symmetry: xx-intercept(s): yy-intercept: a. What is the yy-intercept? Vertex: Axis of symmetry: b. What does the yy-intercept represent? xx-intercept(s): yy-intercept: 147

c. What are the xx-intercepts? We can also use the graph to write the equation of the quadratic function. Recall the standard form of a quadratic equation. d. What do the xx-intercepts represent? ff xx = aaxx & + bbbb + cc There is another form of the quadratic equation called vertex form. e. What is the maximum height of the rocket? Vertex Form: ff(xx) = aa(xx h) & + kk f. When will the rocket reach its maximum height? Ø Ø Ø (h, kk) is the vertex of the graph. aa determines if the graph opens up or down. aa also determines if the parabola is vertically compressed or stretched. g. When is the graph increasing? To write an equation in vertex form from a graph, follow these steps: h. When is the graph decreasing? i. What is the domain of the graph? Step 1: Step 2: Step 3: Substitute the vertex, (h, kk), and the coordinates of another point on the graph, (xx, ff(xx)), into ff(xx) = aa(xx h) & + kk. Solve for aa. Substitute (h, kk) and aa into vertex form. j. What is the range of the graph? 148

2. Recall our graph from exercise 1. Try It! 3. Consider the graph below. a. Substitute the vertex, (h, kk), and the coordinates of another point on the graph, xx, ff xx, into ff(xx) = aa(xx h) & + kk and solve for aa. a. State five observations about the graph. b. Write the function for the graph in vertex form. b. Write the equation of the graph. 149

BEAT THE TEST! 1. The graph of a quadratic function is shown below. Section 6 Topic 2 Nature of the Solutions of Quadratic Equations and Functions Let s use the quadratic formula to discuss the nature of the solutions. Consider the graph of the function ff xx = xx & 4xx + 4. Which statements about this graph are true? Select all that apply. The graph has a yy-intercept at 0, 8. The graph has a maximum point at ( 1, 9). The graph has an xx-intercept at (2, 0). The graph s line of symmetry is the yy-axis. The graph has zeros of 4 and 2. The graph represents the function ff xx = xx 1 & + 9. Where does the parabola intersect the xx-axis? Use the quadratic formula to find the zero(s) of the function. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! 150

Consider the graph of the function ff xx = xx & + 6xx + 8. Consider the graph of the function ff xx = xx & + 6xx 11. Where does the parabola intersect the xx-axis? Where does the parabola intersect the xx-axis? Use the quadratic formula to find the zero(s) of the function. Use the quadratic formula to find the zero(s) of the function. 151

Ø When using the quadratic formula, if the discriminant of the quadratic (the part under the radical) results in a negative number, then the solutions are non-real, complex solutions. Try It! 2. Create a quadratic equation that has complex solutions. Justify your answer. Let s Practice! 1. Use the discriminant to determine if the following quadratic equations have complex or real solution(s). a. 2xx & 3xx 10 = 0 3. Create a quadratic equation that has one real solution. b. xx & 6xx + 9 = 0 c. gg xx = xx & 8xx + 20 152

BEAT THE TEST! 1. Which of the following quadratic equations have real solutions? Select all that apply. 3xx & + 5xx = 11 xx & 12xx + 6 = 0 2xx & + xx + 6 = 0 5xx & 10xx = 3 xx & 2xx = 8 Section 6 Topic 3 Graphing Quadratic Functions Using a Table Suppose you jump into a deep pool of water from a diving platform that is 25 feet above the ground. Your height with respect to time can be modeled by the function HH tt = 25 16tt &, where tt is time in seconds. Complete the table below. Time (seconds) 0 0.25 0.5 0.75 1 1.25 Elevation (feet) Graph function HH(tt) on the following coordinate grid. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! 153

Let s Practice! 1. A construction company builds houses on square-shaped lots of various sizes. The CEO of the company decided to diversify her lots and now has houses built on rectangularshaped lots that are 6 feet longer and 4 feet narrower than her square-shaped lots. a. What is the function that models the size of the rectangular lots relative to the size of the square lots? b. Complete the table below and graph the function. Try It! 2. A business owner recorded the following data for an entire year of sales. Sales Month (in thousands of dollars) Jan 22 Feb 45 Mar 54 April 63 May 70 June 71 July 70 Aug 64 Sept 54 Oct 38 Nov 24 Dec 5 154

a. Plot the data on the graph below. BEAT THE TEST! 1. Consider the following table of values. xx 5 4 3 1 2 4 ff(xx) 16 6 0 0 30 70 Which of the following graphs corresponds to the table of values? A B b. What type of business might be represented by this graph? C D c. Would the quadratic model be an appropriate way to model data for this business going forward? Justify your answer. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! 155

Section 6 Topic 4 Graphing Quadratic Functions Using the Vertex and Intercepts Step 5: Find and plot the xx-intercepts of the function. Factoring is one option, but you can always use the quadratic formula. Given a quadratic equation in standard form, ff(xx) = xx & 4xx 12, use the following steps to graph ff(xx) on the coordinate plane on the following page. Step 1: Use the aa-value to determine if the graph should open upward (positive aa) or downward (negative aa). Graph of ff(xx) = xx & 4xx 12 Step 2: Find and graph the axis of symmetry using the formula xx = A. This is also the h-coordinate of the vertex. &B Step 3: Find ff(h), the kk-coordinate of the vertex, by substituting h into the equation. Plot the vertex, (h, kk), on the graph. Step 4: Find and plot the yy-intercept, which is the constant cc in ff(xx) = aaxx & + bbbb + cc. If possible, use the axis of symmetry to find a reflection point. 156

Let s Practice! 1. Given the function ff(xx) = xx & + 4xx + 21, use the following steps to graph ff(xx) on the coordinate plane on the following page. e. Find and plot the xx-intercepts of the function. Factoring is one option, but you can always use the quadratic formula. a. Use the aa-value to determine if the graph should open upward (positive aa) or downward (negative aa). b. Find and graph the axis of symmetry using the formula xx = CA. This is also the h-coordinate of the vertex. &B Graph of ff xx = xx & + 4xx + 21 c. Find ff(h), the kk-coordinate of the vertex, by substituting h into the equation. Plot the vertex, (h, kk), on the graph. d. Find and plot the yy-intercept, which is the constant cc in ff(xx) = aaxx & + bbbb + cc. If possible, use the axis of symmetry to find a reflection point. 157

Try It! 2. Jorah starts at the top of SlotZilla Zip Line in Las Vegas and rides down Fremont Street. The equation h tt = 2.3tt & + 114 models Jorah s height, in feet, above the ground over time, tt seconds, while he rides the zip line. c. Sketch the graph that models Jorah s height over the time spent riding the zip line. a. What is the vertex of the function h(tt)? b. When will Jorah reach the ground? 158

BEAT THE TEST! 1. On a test, Mia graphed the quadratic function ff xx = xx & 10xx 24. The problem was marked as incorrect. Identify Mia s mistake. Section 6 Topic 5 Graphing Quadratic Functions Using Vertex Form Part 1 Let s review vertex form. Vertex Form: ff(xx) = aa(xx h) & + kk Ø Ø Ø Point (h, kk) is the vertex of the graph. Coefficient aa determines if the graph opens up or down. Coefficient aa also determines if the parabola is vertically stretched or compressed when compared to ff xx = xx &. For example, function ss tt = 16 tt E & + 136, where tt is time & in seconds, models the height of a ball (in feet) that is launched from a balcony of a residential building. Determine and explain whether the graph of the function opens upward or downward. Determine and interpret the coordinates for the vertex of the function. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! Is the function vertically stretched or compressed in comparison to ss tt = tt &? 159

Let s Practice! e. Use the key features to sketch the graph. 1. Given the function ff(xx) = (xx 3) & + 4, use the following steps to graph ff(xx) on the coordinate plane on the following page. a. Use the aa-value to determine if the graph should open upward (positive aa) or downward (negative aa). b. Find and graph the vertex, (h, kk), and axis of symmetry, xx = h. c. Find the yy-intercept by substituting zero for xx. Plot the yy-intercept. If possible, use the axis of symmetry to plot a reflection point. d. Find the xx-intercepts, or zeros, by substituting zero for ff(xx) and solving for xx using square roots. Plot the xx-intercepts. Try It! 2. The yearly profit made by a food truck selling tacos is represented by the following function, where xx represents the number of tacos sold and ff(xx) represents the profit. ff xx = 1 15750 xx 28350 & + 44905 a. The profit function was written in vertex form, ff xx = aa(xx h) & + kk. Examine the values of aa, h, and kk in the profit function above and interpret their meaning(s). 160

b. Graph the profit function on the coordinate plane below. Section 6 Topic 6 Graphing Quadratic Functions Using Vertex Form Part 2 Often times, quadratic equations are not written in vertex form. We can always use the process of completing the square to rewrite quadratic equations in vertex form. Let s Practice! 1. Write the function, ff xx = xx & 4xx 2, in vertex form. Then, graph the function. a. Write the function in standard form. b. Group the quadratic and linear terms together. c. If aa does not equal one, factor aa out of the equation. d. Complete the square. e. Write the function in vertex form. Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! f. Find the zeros, the maximum or minimum point, and the yy-intercept. 161

g. Graph the quadratic, ff xx = xx & 4xx 2, on the coordinate plane below. Try It! 2. Write the function, gg xx = 2xx & 12xx + 17, in vertex form. Then, graph the function. 162

BEAT THE TEST! 1. The graph of gg xx is shown below. 2. Emma rewrote a quadratic function in vertex form. h xx = 4xx & + 16xx + 5 Step 1: h(xx) = 4(xx & + 4xx + ) + 5 + Step 2: h(xx) = 4(xx & + 4xx + 4) + 5 4 Step 3: h xx = 4 xx + 2 & + 1 Part A: Emma said that the vertex is 2, 1. Identify the step where Emma made a mistake, then correct her work. Part B: Does the vertex of h xx represent a maximum or a minimum? Justify your answer. Which function has a maximum that is greater than the maximum of the graph of gg(xx)? A yy = xx 2 & + 4 B yy = xx + 3 & + 2 C yy = F & xx 2 & + 3 D yy = 5 xx + 3 & + 4 Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! 163

Section 6 Topic 7 Transformations of the Dependent Variable of Quadratic Functions Consider the graph and table for the function ff(xx) = xx &. xx ff(xx) 2 4 1 1 0 0 1 1 2 4 Let s Practice! 1. Complete the table to explore what happens when we add a constant to ff xx. xx ff xx gg xx = ff xx + 22 hh xx = ff xx 22 2 4 1 1 0 0 1 1 2 4 Consider the following transformations on the dependent variable ff(xx). gg xx = ff xx + 2 h xx = ff xx 2 mm xx = 2ff(xx) 2. Sketch the graphs of each function on the same coordinate plane with the graph of ff(xx). nn xx = 1 2 ff(xx) pp xx = ff(xx) Why do you think these are called transformations on the dependent variable? 164

Try It! 3. Complete the table to determine what happens when we multiply ff(xx) by a constant. BEAT THE TEST! 1. Given the function ff xx = xx & + 3, identify the effect on the graph of ff(xx) by replacing ff(xx) with: xx ff xx mm xx = 2222(xx) nn xx = 11 22 ff(xx) pp xx = ff(xx) ff xx + kk, where kk > 0. A. Vertically compressed ff(xx) by a factor of kk. 2 4 1 1 0 0 ff xx + kk, where kk < 0. kkkk(xx), where kk > 1. B. C. Shifted ff(xx) down kk units. Reflected ff(xx) about the xx-axis. 1 1 2 4 kkkk(xx), where 0 < kk < 1. D. kkkk xx, where kk = 1. E. Vertically stretched ff(xx) by a factor of kk. Shifted ff(xx) up kk units. 4. Sketch the graphs of each function on the same coordinate plane with the graph of ff(xx). 165

2. The graph of gg(xx) is shown below. Section 6 Topic 8 Transformations of the Independent Variable of Quadratic Functions Consider the graph and table for the function ff(xx) = xx &. xx ff(xx) 2 4 1 1 0 0 1 1 2 4 If ff xx = 3gg xx + 2, identify three ordered pairs that lie on ff xx. Consider the following transformations on the independent variable xx. gg xx = ff xx + 2 h xx = ff xx 2 mm xx = ff(2xx) nn xx = ff 1 2 xx Why do you think these are called transformations on the independent variable? Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! 166

Let s Practice! 1. Complete the table to determine what happens when you add a positive constant to xx. Try It! 3. Complete the table to determine what happens when you add a negative constant to xx. xx ff xx xx gg xx = ff xx + 22 gg(xx) 2 4 4 gg( 4) = ff( 4 + 2) = ff( 2) 4 1 1 3 gg( 3) = ff( 3 + 2) = ff( 1) 1 0 0 1 1 2 4 xx ff xx xx hh xx = ff xx 22 hh(xx) 2 4 0 h(0) = ff(0 2) = ff( 2) 4 1 1 1 h(1) = ff(1 2) = ff( 1) 1 0 0 1 1 2 4 2. Sketch the graph of gg(xx) on the same coordinate plane with the graph of ff(xx). 4. Sketch the graph of h(xx) on the same coordinate plane with the graph of ff(xx). 167

Let s Practice! 5. Complete the table to determine what happens when you multiply xx by a number greater than 1. Try It! 7. Complete the table to determine what happens when you multiply xx by a constant between 0 and 1. xx ff xx xx mm xx = ff 2222 mm(xx) xx ff xx xx nn xx = ff 11 xx nn(xx) 22 2 4 1 mm( 1) = ff(2 1) = ff( 2) 4 2 4 4 nn( 4) = ff 1 4 = ff( 2) 4 2 1 1 1 2 0 0 mm 1 2 = ff 2 1 2 = ff( 1) 1 1 1 2 nn 2 = ff 1 2 = ff( 1) 1 2 0 0 1 1 1 1 2 4 2 4 6. Sketch the graph of mm(xx) on the same coordinate plane with the graph of ff(xx). 8. Sketch the graph of nn(xx) on the same coordinate plane with the graph of ff(xx). 168

BEAT THE TEST! 1. The table that represents the quadratic function gg(xx) is shown below. xx gg(xx) 6 12 4 2 1 12 7 90 11 182 Section 6 Topic 9 Finding Solution Sets to Systems of Equations Using Tables of Values and Successive Approximations We can find solutions to systems of linear and quadratic equations by looking at a graph or table. Consider the following system of equations. ff xx = xx & + 5xx + 6 gg xx = 2xx + 6 The graph of the system is shown below. The function ff xx = gg F xx. Complete the following table E for ff xx. xx ff(xx) Algebra Wall Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started! For which values of xx does ff xx = gg(xx)? We call these the solutions of ff xx = gg xx. 169

We can also identify the solutions by looking at tables. We can easily find the solutions by looking for the xx-coordinate where ff xx = gg xx. The table that represents the system is shown below. xx ff(xx) gg(xx) 3 0 0 2 0 2 1 2 4 0 6 6 1 12 8 2 20 10 3 30 12 Use the table to identify the solutions of ff xx = gg(xx). We can also use a process called successive approximations. Consider the following system. ff xx = xx & + 2xx + 1 gg xx = 2xx + 3 The table that represents the system is shown below. xx ff(xx) gg(xx) 0 1 3 0.5 2.25 4 1 4 5 1.5 6.25 6 2 9 7 2.5 12.25 8 3 16 9 Since there are no xx-coordinates where ff xx = gg(xx), we must look for the xx-coordinates that have the smallest absolute differences in ff(xx) and gg xx. Ø Ø Ø Find the absolute differences in ff(xx) and gg(xx) on the table above. Between which two xx values must the positive solution lie? Which of the values does the solution lie closest to? 170

Let s Practice! 1. Using the same system, complete the table below. Try It! 4. The graphs of ff(xx) and gg(xx) are shown below. ff xx = xx & + 2xx + 1 gg xx = 2xx + 3 xx ff(xx) gg(xx) 1 4 5 1.1 4.41 5.2 1.2 5.4 1.3 5.29 1.4 1.5 6.25 6 2. Find the absolute differences in ff(xx) and gg(xx) on the table above. 3. Use the table to find the positive solution (to the nearest tenth) for ff xx = gg xx. Use the graph to find the negative and positive solution of ff xx = gg(xx). 171

BEAT THE TEST! 1. Consider the following system of equations. gg xx = xx & 10 h xx = xx + 8 THIS PAGE WAS INTENTIONALLY LEFT BLANK The table below represents the system. xx gg(xx) hh(xx) 4 6 4 3.5 2.25 4.5 3 1 5 2.5 3.75 5.5 2 6 6 1.5 7.75 6.5 1 9 7 Use successive approximations to find the negative solution for gg xx = h(xx). Test Yourself! Practice Tool Great job! You have reached the end of this section. Now it s time to try the Test Yourself! Practice Tool, where you can practice all the skills and concepts you learned in this section. Log in to Algebra Nation and try out the Test Yourself! Practice Tool so you can see how well you know these topics! 172