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Section 6: Quadratic Equations and Functions –Part 2Section 6 – Topic 1Observations from a Graph of a Quadratic FunctionLet’s review some things we learned earlier about theinformation we can gather from the graph of a quadraticfunction.Vertex:(𝟏, 𝟏)Axis of symmetry:𝒙 𝟏𝑥-intercept(s):(𝟎, 𝟎) and (𝟐, 𝟎)𝑦-intercept:(𝟎, 𝟎)Vertex:( 𝟏, 𝟏)Axis of symmetry:𝒙 𝟏𝑥-intercept(s):none𝑦-intercept:(𝟎, 𝟐)
Let’s Practice!1. The graph shows the height of a rocket from the time itwas launched from the ground. Use the graph to answerthe questions below.a. What is the 𝑦-intercept?(𝟎, 𝟎)b. What does the 𝑦-intercept represent?The point from which the rocket was launched.
c. What are the 𝑥-intercepts?(𝟎, 𝟎) and (𝟏𝟐, 𝟎)d. What do the 𝑥-intercepts represent?The time the rocket was launched and the number ofseconds it took for the rocket to hit the ground.e. What is the maximum height of the rocket?𝟔𝟐𝟓 feetf.When will the rocket reach its maximum height?𝟔 secondsg. When is the graph increasing?Set Notation 𝟎 𝒙 𝟔Interval Notation 𝟎, 𝟔h. When is the graph decreasing?Set Notation 𝟔 𝒙 𝟏𝟐Interval Notation 𝟔, 𝟏𝟐i.What is the domain of the graph?Set Notation 𝒙 𝟎 𝒙 𝟏𝟐Interval Notation 𝟎, 𝟏𝟐j.What is the range of the graph?Set Notation 𝒚 𝟎 𝒚 𝟔𝟐𝟓Interval Notation 𝟎, 𝟔𝟐𝟓
We can also use the graph to write the equation of thequadratic function.Recall the standard form of a quadratic equation.𝑓 𝑥 𝑎𝑥 3 𝑏𝑥 𝑐There is another form of the quadratic equation called vertexform.Vertex Form: 𝑓(𝑥) 𝑎(𝑥 ℎ)3 𝑘Ø (ℎ, 𝑘) is the vertex of the graph.Ø 𝑎 determines if the graph opens up or down.Ø 𝑎 also determines if the parabola is verticallycompressed or stretched.To write an equation in vertex form from a graph, follow thesesteps:Step 1:Substitute the vertex, (ℎ, 𝑘), and the coordinatesof another point on the graph, (𝑥, 𝑓(𝑥)), into𝑓(𝑥) 𝑎(𝑥 ℎ)3 𝑘.Step 2:Solve for 𝑎.Step 3:Substitute (ℎ, 𝑘) and 𝑎 into vertex form.
2. Recall our graph from exercise 1.a. Substitute the vertex, (ℎ, 𝑘), and the coordinates ofanother point on the graph,𝑥, 𝑓 𝑥 , into𝑓(𝑥) 𝑎(𝑥 ℎ)3 𝑘 and solve for 𝑎.𝒉, 𝒌 (𝟔, 𝟔𝟐𝟓) Coordinate: (𝟎, 𝟎)𝟎 𝒂 𝟎 𝟔 𝟐 𝟔𝟐𝟓𝟔𝟐𝟓 𝟑𝟔𝒂 𝟑𝟔𝟑𝟔𝟔𝟐𝟓𝒂 𝟏𝟕. 𝟑𝟔𝟑𝟔b. Write the function for the graph in vertex form.𝒇 𝒙 𝟔𝟐𝟓𝟑𝟔𝒙 𝟔𝟐 𝟔𝟐𝟓or 𝒇 𝒙 𝟏𝟕. 𝟑𝟔 𝒙 𝟔𝟐 𝟔𝟐𝟓
Try It!3. Consider the graph below.a. State five observations about the graph.Vertex: ( 𝟑, 𝟒)𝒙-intercepts ( 𝟏, 𝟎) and ( 𝟓, 𝟎)𝒚-intercept (𝟎, 𝟓)Domain: 𝒙 𝒙 𝓡 or , Range: 𝒚 𝒚 𝟒 or – 𝟒, )Models a quadratic where 𝒂 is positive.b. Write the equation of the graph.𝒇 𝒙 𝒂 𝒙 𝒉 𝟐 𝒌𝒉, 𝒌 ( 𝟑, 𝟒) Coordinate: ( 𝟏, 𝟎)𝟎 𝒂 𝟏 𝟑 𝟐 𝟒𝟎 𝟒𝒂 𝟒𝒂 𝟏𝒇 𝒙 𝒙 𝟑 𝟐 𝟒
BEAT THE TEST!1. The graph of a quadratic function is shown below.Which statements about this graph are true? Select allthat apply.ý ý ý ý The graph has a 𝑦-intercept at 0, 8 .The graph has a relative maximum at ( 1, 9).The graph has an 𝑥-intercept at (2, 0).The line of symmetry is the 𝑦-axis.The graph has zeros of 4 and 2.The graph represents the function 𝑓 𝑥 𝑥 13 9.AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 2Nature of the Zeros of Quadratic Equations andFunctionsLet’s use the quadratic formula to discuss the nature of thezero(s) of a quadratic function.Consider the graph of the function 𝑓 𝑥 𝑥 3 4𝑥 4.Where does the parabola intersect the 𝑥-axis?At (𝟐, 𝟎)Use the quadratic formula to find the zero(s) of the function.𝒙 𝟒 𝟒 𝟐 𝟒(𝟏)(𝟒) 𝟒 𝟏𝟔 𝟏𝟔 𝟒 𝟎 𝟒 𝟐𝟐(𝟏)𝟐𝟐𝟐
Consider the graph of the function 𝑓 𝑥 𝑥 3 6𝑥 8.Where does the parabola intersect the 𝑥-axis?At ( 𝟐, 𝟎) and ( 𝟒, 𝟎)Use the quadratic formula to find the zero(s) of the function.𝒙 𝒙 𝟔 𝟔 𝟐 𝟒(𝟏)(𝟖) 𝟔 𝟑𝟔 𝟑𝟐 𝟔 𝟒 𝟔 𝟐 𝟐(𝟏)𝟐𝟐𝟐 𝟔 𝟐𝟒 𝟔 𝟐𝟖 𝟐or 𝒙 𝟐𝟐𝟐𝟐 𝟒
Consider the graph of the function 𝑓 𝑥 𝑥 3 6𝑥 11.Where does the parabola intersect the 𝑥-axis?It doesn’t intersect the 𝒙-axis.Use the quadratic formula to find the zero(s) of the function.𝒙 𝟔 𝟔no real zeros𝟐 𝟒( 𝟏)( 𝟏𝟏) 𝟔 𝟑𝟔 𝟒𝟒 𝟔 𝟖 𝟐( 𝟏) 𝟐 𝟐
Ø When using the quadratic formula, if thediscriminant of the quadratic function (thepart under the radical) is a negative number,then the zeros are non-real, complexsolutions.Let’s Practice!1. Use the discriminant to determine if the followingquadratic equations have complex or real zero(s).a. 2𝑥 3 3𝑥 10 0 𝟑 𝟐 𝟒(𝟐)( 𝟏𝟎) 𝟗 𝟖𝟎 𝟖𝟗The equation has two real zeros.b. 𝑥 3 6𝑥 9 0 𝟔 𝟐 𝟒(𝟏)(𝟗) 𝟑𝟔 𝟑𝟔 𝟎The equation has one real zero.c. 𝑔 𝑥 𝑥 3 8𝑥 20 𝟖 𝟐 𝟒(𝟏)(𝟐𝟎) 𝟔𝟒 𝟖𝟎 𝟏𝟔The equation has complex zeros.
Try It!2. Create a quadratic equation that has complex zeros.Justify your answer.Answers vary.Sample answer: 𝒂 𝟏𝟎, 𝒃 𝟑, 𝒄 𝟐𝒇 𝒙 𝟏𝟎𝒙𝟐 𝟑𝒙 𝟐 𝟑𝟐 𝟒(𝟏𝟎)(𝟐) 𝟗 𝟖𝟎 𝟕𝟏3. Create a quadratic equation that has one real zero.Answers vary (Any perfect square trinomial)Sample answer: 𝒂 𝟏, 𝒃 𝟏𝟖, 𝒄 𝟖𝟏𝒇 𝒙 𝒙𝟐 𝟏𝟖𝒙 𝟖𝟏𝟏𝟖𝟐 𝟒(𝟏)(𝟖𝟏) 𝟑𝟐𝟒 𝟑𝟐𝟒 𝟎
BEAT THE TEST!1. Which of the following quadratic equations have realzeros? Select all that apply. ý ý 𝑓𝑓𝑓𝑓𝑓𝑥𝑥𝑥𝑥𝑥 3𝑥 3 5𝑥 11 𝑥 3 12𝑥 6 2𝑥 3 𝑥 6 5𝑥 3 10𝑥 3 𝑥 3 2𝑥 8AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 3Graphing Quadratics Using a TableSuppose you jump into a deep pool of water from a divingplatform that is 25 feet above the ground. Your height withrespect to time can be modeled by the function𝐻 𝑡 25 16𝑡 3 , where 𝑡 is time in seconds.Complete the table below.Time (seconds)00.250.50.7511.25Elevation ation (feet)Graph function 𝐻(𝑡) on the following coordinate grid.Time (Seconds)
Let’s Practice!1. A construction company builds houses on square-shapedlots of various sizes. The CEO of the company decided todiversify her lots and now has houses built on rectangularshaped lots that are 6 feet longer and 4 feet narrowerthan her square-shaped lots.a. What is the function that models the size of therectangular lots relative to the size of the square lots?𝒇 𝒙 𝒙 𝟔 𝒙 𝟒 𝒙𝟐 𝟐𝒙 e Lot Sizeb. Complete the table below and graph the function.Square Lot Size
Try It!2. A business owner recorded the following data for anentire year of sales.SalesMonth(in thousandsof 2455463707170645438245
Salesa. Plot the data on the graph below.Month of Yearb. What type of business might be represented by thisgraph?Answers Vary.Sample answers: Shaved Ice Cones or Theme Parkc. Would the quadratic model be an appropriate wayto model data for this business going forward? Justifyyour answer.Yes, it follows a parabolic pattern.
BEAT THE TEST!1. Consider the following table of values.𝒙 5 4 3 124𝒇(𝒙) 16 600 30 70Which of the following is the graph corresponding to thetable of values?ABCDAnswer: DAlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 4Graphing Quadratics Using the Vertex and InterceptsGiven a quadratic equation in standard form,𝑓(𝑥) 𝑥 3 4𝑥 12, use the following steps to graph 𝑓(𝑥) onthe coordinate plane on the following page.Step 1: Use the 𝑎-value to determine if the graph should openupward (positive 𝑎) or downward (negative 𝑎).𝒂 𝟏, Opens upwardStep 2: Find and graph the axis of symmetry using the formulaX𝑥 . This is also the ℎ-coordinate of the vertex.3Y𝒙 𝟒𝟒 𝟐𝟐 𝟏𝟐Step 3: Find 𝑓(ℎ), the 𝑘-coordinate of the vertex, bysubstituting ℎ into the equation. Plot the vertex, (ℎ, 𝑘),on the graph.𝒇 𝟐 𝟐𝟐 𝟒 𝟐 𝟏𝟐 𝟒 𝟖 𝟏𝟐 𝟏𝟔𝒉, 𝒌 𝟐, 𝟏𝟔Step 4: Find and plot the 𝑦-intercept, which is the constant 𝑐in 𝑓(𝑥) 𝑎𝑥 3 𝑏𝑥 𝑐. If needed, use the axis ofsymmetry to find a reflection point.𝒚-intercept: (𝟎, 𝟏𝟐)
Step 5: Find and plot the 𝑥-intercepts of the function.Factoring is one option, but you can always use thequadratic formula.𝒙𝟐 𝟒𝒙 𝟏𝟐 𝟎𝒙 𝟔 𝒙 𝟐 𝟎𝒙 𝟔 or 𝒙 𝟐Graph of 𝑓(𝑥) 𝑥 3 4𝑥 12
Let’s Practice!1. Given the function 𝑓(𝑥) 𝑥 3 4𝑥 21, use the followingsteps to graph 𝑓(𝑥) on the coordinate plane on thefollowing page.a. Use the 𝑎-value to determine if the graph should openupward (positive 𝑎) or downward (negative 𝑎).𝒂 𝟏, Opens downwardb. Find and graph the axis of symmetry using the formula𝑥 ZX3Y𝒙 . This is also the ℎ-coordinate of the vertex.𝟒𝟒 𝟐𝟐 𝟏𝟐c. Find 𝑓(ℎ), the 𝑘-coordinate of the vertex, bysubstituting ℎ into the equation. Plot the vertex, (ℎ, 𝑘),on the graph.𝒇 𝟐 𝟐𝟐 𝟒 𝟐 𝟐𝟏 𝟒 𝟖 𝟐𝟏 𝟐𝟓𝒉, 𝒌 𝟐, 𝟐𝟓d. Find and plot the 𝑦-intercept, which is the constant 𝑐in 𝑓(𝑥) 𝑎𝑥 3 𝑏𝑥 𝑐. If possible, use the axis ofsymmetry to find a reflection point.𝒚-intercept: (𝟎, 𝟐𝟏)
e. Find and plot the 𝑥-intercepts of the function.Factoring is one option, but you can always use thequadratic formula. 𝒙𝟐 𝟒𝒙 𝟐𝟏 𝟎𝒙𝟐 𝟒𝒙 𝟐𝟏 𝟎𝒙 𝟕 𝒙 𝟑 𝟎𝒙 𝟕 or 𝒙 𝟑Graph of 𝑓 𝑥 𝑥 3 4𝑥 21
Try It!2. Jorah starts at the top of SlotZilla Zip Line in Las Vegas andrides down Fremont Street. The equationℎ 𝑡 2.3𝑡 3 114 models Jorah’s height, in feet, abovethe ground over time, 𝑡 seconds, while he rides the zip line.a. What is the vertex of the function ℎ(𝑡)?𝒙 𝟎 𝟎𝟐 𝟐. 𝟑𝒉 𝟎 𝟐. 𝟑 𝟎𝟐 𝟏𝟏𝟒 𝟏𝟏𝟒𝒉, 𝒌 (𝟎, 𝟏𝟏𝟒)b. When will Jorah reach the ground? 𝟐. 𝟑𝒕𝟐 𝟏𝟏𝟒 𝟎 𝟐. 𝟑𝒕𝟐 𝟏𝟏𝟒 𝟏𝟏𝟒𝒕𝟐 𝟐. 𝟑𝒕 𝟏𝟏𝟒𝟐.𝟑, 𝒕 𝟕. 𝟎𝟒It will take Jorah about 𝟕 seconds to reach theground.
Height in feetc. Sketch the graph that models Jorah’s height over thetime spent riding the zip line.Time in seconds
BEAT THE TEST!1. On a test, Mia graphed the quadratic function𝑓 𝑥 𝑥 3 10𝑥 24. The problem was marked asincorrect. Identify Mia’s mistake.Mia’s 𝒚 intercept is not correct.Mia’s zeros are not correct.𝒙𝟐 𝟏𝟎𝒙 𝟐𝟒 𝟎(𝒙 𝟏𝟐)(𝒙 𝟐) 𝟎𝒙 𝟏𝟐 or 𝒙 𝟐AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 5Graphing Quadratics Using Vertex Form – Part 1Let’s review vertex form.Vertex Form: 𝑓(𝑥) 𝑎(𝑥 ℎ)3 𝑘Ø Point (ℎ, 𝑘) is the vertex of the graph.Ø Coefficient 𝑎 determines if the graph opens up ordown.Ø Coefficient 𝑎 also determines if the parabola isvertically stretched or compressed when comparedto 𝑓 𝑥 𝑥 3 .For example, function 𝑠 𝑡 16 𝑡 33 136, where 𝑡 is timein seconds, models the height of a ball (in feet) that islaunched from a balcony of a residential building.Determine and explain whether the graph of the functionopens upward or downward.Opens downwardDetermine and interpret the coordinates for the vertex of thefunction.𝟑, 𝟏𝟑𝟔𝟐Is the function vertically stretched or compressed incomparison to 𝑠 𝑡 𝑡 3 ?The function is vertically stretched.
Let’s Practice!1. Given the function 𝑓(𝑥) (𝑥 3)3 4, use the followingsteps to graph 𝑓(𝑥) on the coordinate plane on thefollowing page.a. Use the 𝑎-value to determine if the graph should openupward (positive 𝑎) or downward (negative 𝑎).𝒂 𝟏, opens upwardb. Find and graph the vertex, (ℎ, 𝑘), and axis ofsymmetry, 𝑥 ℎ.Vertex: (𝟑, 𝟒)Axis of symmetry: 𝒙 𝟑c. Find the 𝑦-intercept by substituting zero for 𝑥. Plot the𝑦-intercept. If possible, use the axis of symmetry to plota reflection point.𝒚-intercept:(𝟎, 𝟏𝟑)d. Find the 𝑥-intercepts, or zeros, by substituting zero for𝑓(𝑥) and solving for 𝑥 using square roots. Plot the𝑥-intercepts.𝒙 𝟑 𝟐 𝟒 𝟎𝒙 𝟑 𝟐 𝟒𝒙 𝟑 𝟐 𝟒There are no real zeros.
e. Use the key features to sketch the graph.Try It!2. The yearly profit made by a food truck selling tacos isrepresented by the following function, where 𝑥representsthe number of tacos sold and 𝑓(𝑥) represents the profit.𝑓 𝑥 1𝑥 28350157503 44905a. The profit function was written in vertex form,𝑓 𝑥 𝑎(𝑥 ℎ)3 𝑘. Examine the values of 𝑎, ℎ,and 𝑘in the profit function above and interpret theirmeaning(s).𝟏𝒂 , making more profit until a certain # is sold𝟏𝟓𝟕𝟓𝟎and then making less profit. 𝒉, 𝒌 (𝟐𝟖𝟑𝟓𝟎, 𝟒𝟒𝟗𝟎𝟓)maximum profit will be 44,905 when 28,350 tacos aresold.
b. Graph the profit function on the coordinate planebelow.AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 6Graphing Quadratics Using Vertex Form – Part 2Oftentimes, quadratic equations are not written in vertex form.We can always use the process of completing the square torewrite quadratic equations in vertex form.Let’s Practice!1. Write the function, 𝑓 𝑥 𝑥 3 4𝑥 2, in vertex form. Then,graph the function.a. Write the function in standard form.𝒇 𝒙 𝒙𝟐 𝟒𝒙 𝟐b. Group the quadratic and linear term together.𝒇 𝒙 (𝒙𝟐 𝟒𝒙) 𝟐c. If 𝑎 does not equal one, factor 𝑎 out of the equation.𝒂 𝟏d. Complete the square.(𝒙𝟐 𝟒𝒙 𝟒) 𝟐 𝟒e. Write the function in vertex form.𝒇 𝒙 𝒙 𝟐 𝟐 𝟔f.Find the zeros, the maximum or minimum point, andthe 𝑦-intercept.Zeros: 𝒙 𝟐 𝟔 𝟒. 𝟒Vertex: (𝟐, 𝟔)𝒙 𝟐 𝟔 𝟎. 𝟓𝒚-interecept (𝟎, 𝟐)
g. Graph the quadratic, 𝑓 𝑥 𝑥 3 4𝑥 2,on thecoordinate plane below.
Try It!2. Write the function, 𝑔 𝑥 2𝑥 3 12𝑥 17, in vertex form.Then, graph the function.𝒈 𝒙 𝟐𝒙𝟐 𝟏𝟐𝒙 𝟏𝟕𝒈 𝒙 𝟐(𝒙𝟐 𝟔𝒙 𝟗) 𝟏𝟕 𝟏𝟖𝒈 𝒙 𝟐 𝒙 𝟑 𝟐 𝟏Graph of 𝑔 𝑥 2𝑥 3 12𝑥 17
BEAT THE TEST!1. The graph of 𝑔 𝑥 is shown below.Which function has a maximum that is greater than themaximum of the graph of 𝑔(𝑥)?ABCD𝑦 𝑥 2 3 4𝑦 𝑥 3 3 2 𝑦 𝑥 2 3 33𝒚 𝟓 𝒙 𝟑Answer: D𝟐 𝟒
2. Emma rewrote a quadratic function in vertex form.ℎ 𝑥 4𝑥 3 16𝑥 5Step 1:ℎ(𝑥) 4(𝑥 3 4𝑥 ) 5Step 2: ℎ(𝑥) 4(𝑥 3 4𝑥 4) 5 4Step 3: ℎ 𝑥 4 𝑥 2 3 1Part A: Emma said that the vertex is 2, 1 . Identify thestep where Emma made a mistake, then correcther work.Velma is not correct. Her mistake is in step 𝟐. Sheshould have subtracted 𝟏𝟔 instead of 𝟒.𝟒 𝒙𝟐 𝟒𝒙 𝟒 𝟓 𝟏𝟔𝟒 𝒙 𝟐 𝟐 𝟏𝟏The vertex should be ( 𝟐, 𝟏𝟏)Part B: Does the vertex of ℎ 𝑥 represent a maximum or aminimum? Justify your answer.The vertex represents a minimum, because the value of 𝒂is positive ( 4).AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 7Transformations of the Dependent Variable of QuadraticFunctionsConsider the graph and table for the function 𝑓(𝑥) 𝑥 3 .𝒙𝒇(𝒙) 24 11001124Consider the following transformations on the dependentvariable 𝑓(𝑥).𝑔 𝑥 𝑓 𝑥 2ℎ 𝑥 𝑓 𝑥 2𝑚 𝑥 2𝑓(𝑥)1𝑛 𝑥 𝑓(𝑥)2𝑝 𝑥 𝑓(𝑥)Why do you think these are called transformations on thedependent variable?Because 𝒇(𝒙) represents the dependent variable and in eachfunction we are performing an operation on 𝒇 𝒙 .
Let’s Practice!1. Complete the table to explore what happens when weadd a constant to 𝑓 𝑥 .𝒙𝒇 𝒙𝒈 𝒙 𝒇 𝒙 𝟐𝒉 𝒙 𝒇 𝒙 𝟐 24𝟔𝟐 11𝟑 𝟏00𝟐 𝟐11𝟑 𝟏24𝟔𝟐2. Sketch the graphs of each function on the samecoordinate plane with the graph of 𝑓(𝑥).
Try It!3. Complete the table to determine what happens when wemultiply 𝑓(𝑥) by a constant.𝒎 𝒙 𝟐𝒇(𝒙) 𝒏 𝒙 𝟏𝒇(𝒙) 𝒑 𝒙 𝒇(𝒙)𝟐𝒙𝒇 𝒙 24𝟖𝟐 𝟒 11𝟐𝟏𝟐 𝟏00𝟎𝟎𝟎11𝟐𝟏𝟐 𝟏24𝟖𝟐 𝟒4. Sketch the graphs of each function on the samecoordinate plane with the graph of 𝑓(𝑥).
BEAT THE TEST!1. Given the function 𝑓 𝑥 𝑥 3 3,identify the effect on thegraph of 𝑓(𝑥) by replacing 𝑓(𝑥) with:E𝑓 𝑥 𝑘, where 𝑘 0.B𝑓 𝑥 𝑘, where 𝑘 0.D𝑘𝑓(𝑥), where 𝑘 1.A𝑘𝑓(𝑥), where 0 𝑘 1.C𝑘𝑓 𝑥 , where 𝑘 1.A.Vertically compressed𝑓(𝑥) by a factor of 𝑘.B.Shifted 𝑓(𝑥) down 𝑘units.C.Reflected 𝑓(𝑥) aboutthe 𝑥-axis.D.Vertically stretched𝑓(𝑥) by a factor of 𝑘.E.Shifted 𝑓(𝑥) up 𝑘units.
2. The graph of 𝑔(𝑥) is shown below.If 𝑓 𝑥 3𝑔 𝑥 2, identify three ordered pairs that lie on𝑓 𝑥 .Answers vary. Consider the following three points on 𝒈 𝒙 .𝟎, 𝟑 , 𝟏, 𝟐 ,and 𝟐, 𝟑 .𝒇 𝟎 𝟑 𝒈 𝟎 𝟐 𝟑 𝟑 𝟐 𝟏𝟏𝒇 𝟏 𝟑 𝒈 𝟏 𝟐 𝟑 𝟐 𝟐 𝟖𝒇 𝟐 𝟑 𝒈 𝟐 𝟐 𝟑 𝟑 𝟐 𝟏𝟏Three points on 𝒇(𝒙) would be 𝟎, 𝟏𝟏 , 𝟏, 𝟖 , and 𝟐, 𝟏𝟏 .AlgebraWallWant some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 8Transformations of the Independent Variable ofQuadratic FunctionsConsider the graph and table for the function 𝑓(𝑥) 𝑥 3 .𝒙𝒇(𝒙) 24 11001124Consider the following transformations on the independentvariable 𝑥.𝑔 𝑥 𝑓 𝑥 2ℎ 𝑥 𝑓 𝑥 2𝑚 𝑥 𝑓(2𝑥)𝑛 𝑥 𝑓1𝑥2Why do you think these are called transformations on theindependent variable?Because 𝒙 represents the independent variable and in eachfunction we are performing an operation on 𝒙.
Let’s Practice!1. Complete the table to determine what happens whenyou add a positive constant to 𝑥.𝒙𝒇 𝒙𝒙𝒈 𝒙 𝒇 𝒙 𝟐𝒈(𝒙) 24 4𝑔( 4) 𝑓( 4 2) 𝑓( 2)4 11 3𝑔( 3) 𝑓( 3 2) 𝑓( 1)100 𝟐𝒈( 𝟐) 𝒇( 𝟐 𝟐) 𝒇(𝟎)𝟎11 𝟏𝒈( 𝟏) 𝒇( 𝟏 𝟐) 𝒇(𝟏)𝟏24𝟎𝒈(𝟎) 𝒇(𝟎 𝟐) 𝒇(𝟐)𝟒2. Sketch the graph of 𝑔(𝑥) on the same coordinate planewith the graph of 𝑓(𝑥).
Try It!3. Complete the table to determine what happens whenyou add a negative constant to 𝑥.𝒙𝒇 𝒙𝒙𝒉 𝒙 𝒇 𝒙 𝟐𝒉(𝒙) 240ℎ(0) 𝑓(0 2) 𝑓( 2)4 111ℎ(1) 𝑓(1 2) 𝑓( 1)100𝟐𝒉(𝟐) 𝒇(𝟐 𝟐) 𝒇(𝟎)𝟎11𝟑𝒉(𝟑) 𝒇(𝟑 𝟐) 𝒇(𝟏)𝟏24𝟒𝒉(𝟒) 𝒇(𝟒 𝟐) 𝒇(𝟐)𝟒4. Sketch the graph ofℎ(𝑥) on the same coordinate planewith the graph of 𝑓(𝑥).
Let’s Practice!5. Complete the table to determine what happens whenyou multiply 𝑥 by a number greater than 1.𝒙𝒇 𝒙𝒙𝒎 𝒙 𝒇 𝟐𝒙 24 1𝑚( 1) 𝑓(2 1) 𝑓( 2) 111 2𝑚 00𝟎𝒎(𝟎) 𝒇(𝟐 𝟎) 𝒇(𝟎)𝟎11𝟏𝟐𝟏𝟏𝒎 𝒇 𝟐 𝒇(𝟏)𝟐𝟐𝟏24𝟏𝒎(𝟏) 𝒇(𝟐 𝟏) 𝒇(𝟐)𝟒11 𝑓 2 𝑓( 1)22𝒎(𝒙)416. Sketch the graph of 𝑚(𝑥) on the same coordinate planewith the graph of 𝑓(𝑥).
Try It!7. Complete the table to determine what happens whenyou multiply 𝑥 by a constant between 0and 1.𝒙𝒇 𝒙𝒙 24 4 11 200𝟎11𝟐24𝟒𝒏 𝒙 𝒇𝟏𝒙𝟐1𝑛( 4) 𝑓( 4) 𝑓( 2)21𝑛 2 𝑓 2 𝑓( 1)2𝟏𝒏(𝟎) 𝒇( 𝟎) 𝒇(𝟎)𝟐𝟏𝒏 𝟐 𝒇 𝟐 𝒇(𝟏)𝟐𝟏𝒏(𝟏) 𝒇 𝟒 𝒇(𝟐)𝟐𝒏(𝒙)41𝟎𝟏𝟒8. Sketch the graph of 𝑛(𝑥) on the same coordinate planewith the graph of 𝑓(𝑥).
BEAT THE TEST!1. The table that represents the quadratic function 𝑔(𝑥) isshown below.𝒙𝒈(𝒙) 612 4211279011182The function 𝑓 𝑥 𝑔for 𝑓 𝑥 .𝒙𝒇(𝒙) 𝟏𝟖𝟏𝟐 ��𝟖𝟐AlgebraWall 𝑥 . Complete the following table𝟏 𝟏𝟖 𝒈 𝟔 𝟏𝟐𝟑𝟏𝒇 𝟏𝟐 𝒈 𝟏𝟐 𝒈 𝟒 𝟐𝟑𝟏𝒇 𝟑 𝒈 𝟑 𝒈 𝟏 𝟏𝟐𝟑𝟏𝒇 𝟐𝟏 𝒈 𝟐𝟏 𝒈 𝟕 𝟗𝟎𝟑𝟏𝒇 𝟑𝟑 𝒈 𝟑𝟑 𝒈 𝟏𝟏 𝟏𝟖𝟐𝟑𝒇 𝟏𝟖 𝒈Want some help? You can always ask questions onthe Algebra Wall and receive help from otherstudents, teachers, and Study Experts. You can alsohelp others on the Algebra Wall and earn KarmaPoints for doing so. Go to AlgebraNation.com to learnmore and get started!
Section 6 – Topic 9Finding Solution Sets to Systems of Equations UsingTables of Values and Successive ApproximationsWe can find solutions to systems of linear and quadraticequations by looking at a graph or table.Consider the following system of equations.𝑓 𝑥 𝑥 3 5𝑥 6𝑔 𝑥 2𝑥 6The graph of the system is shown below.For which values of 𝑥 does 𝑓 𝑥 𝑔(𝑥)? 𝒙 𝟑 or 𝒙 𝟎We call these the solutions of 𝑓 𝑥 𝑔 𝑥 .
We can also identify the solutions by looking at tables. We caneasily find the solutions by looking for the 𝑥-coordinate where𝑓 𝑥 𝑔 𝑥 .The table that represents the system is shown below.𝒙𝒇(𝒙)𝒈(𝒙) 300 202 12406611282201033012Use the table to identify the solutions of 𝑓 𝑥 𝑔(𝑥).𝒇 𝒙 and 𝒈(𝒙) are equat at 𝒙 𝟑 and 𝒙 𝟎.
We can also use a process called successive approximations.Consider the following system.𝑓 𝑥 𝑥 3 2𝑥 1𝑔 𝑥 2𝑥 3The table that represents the systems is shown below.𝒙𝒇(𝒙)𝒈(𝒙)013𝟐0.52.254𝟏. 𝟕𝟓145𝟏1.56.256𝟎. 𝟐𝟓297𝟐2.512.258𝟒. 𝟐𝟓3169𝟕Since there are no 𝑥-coordinates where 𝑓 𝑥 𝑔(𝑥), we mustlook for the 𝑥-coordinates that have the smallest absolutedifferences in 𝑓(𝑥) and 𝑔 𝑥 .Ø Find the absolute differences in 𝑓(𝑥) and 𝑔(𝑥) on thetable above.Ø Between which two 𝑥values must the positive solutionlie? 𝟏 and 𝟏. 𝟓Ø Which of the values does the solution lie closest to?𝟏. 𝟓
Let’s Practice!1. Using the same system, complete the table below.𝑓 𝑥 𝑥 3 2𝑥 1𝑔 𝑥 2𝑥 3𝒙𝒇(𝒙)𝒈(𝒙)145𝟏1.14.415.2𝟎. 𝟕𝟗1.2𝟒. 𝟖𝟒5.4𝟎. 𝟓𝟔1.35.29𝟓. 𝟔𝟎. 𝟑𝟏1.4𝟓. 𝟕𝟔𝟓. 𝟖𝟎. 𝟎𝟒1.56.256𝟎. 𝟐𝟓2. Find the absolute differences in 𝑓(𝑥) and 𝑔(𝑥) on the tableabove.3. Use the table to find the positive solution (to the nearesttenth) for 𝑓 𝑥 𝑔 𝑥 .𝒙 𝟏. 𝟒
Try It!4. The graphs of 𝑓(𝑥) and 𝑔(𝑥) are shown below.Use the graph to find the negative and positive solution of𝑓 𝑥 𝑔(𝑥).𝒙 𝟏𝒙 𝟒
BEAT THE TEST!1. Consider the following system of equations.𝑔 𝑥 𝑥 3 10ℎ 𝑥 𝑥 8The table below represents the system.𝒙𝒈(𝒙)𝒉(𝒙) 464𝟐 3.52.254.5𝟐. 𝟐𝟓 3 15𝟔 2.5 3.755.5𝟗. 𝟐𝟓 2 66𝟏𝟐 1.5 7.756.5𝟏𝟒. 𝟐𝟓 1 97𝟏𝟔Use successive approximations to find the negativesolution for 𝑔 𝑥 ℎ(𝑥).𝒈 𝟑. 𝟗 𝟓. 𝟐𝟏𝒈 𝟑. 𝟖 𝟒. 𝟒𝟒𝒈 𝟑. 𝟕 𝟑. 𝟔𝟗𝒉 𝟑. 𝟗 𝟒. 𝟏𝒉 𝟑. 𝟖 𝟒. 𝟐𝒉 𝟑. 𝟕 𝟒. 𝟑Difference: 𝟏. 𝟏𝟏Difference: 𝟎. 𝟐𝟒Difference: 𝟎. 𝟔𝟏The negative solution to the nearest tenth is 𝒙 𝟑. 𝟖Now it’s time to try the “Test Yourself! Practice Tool,”where you can practice all the skills and conceptsyou learned in this section. Log in to Algebra Nationand try out the “Test Yourself! Practice Tool” so youTest Yourself! can see how well you know these topics!Practice Tool
We can also use the graph to write the equation of the quadratic function. Recall the standard form of a quadratic equation. 1( 2(3 5( 6 There is another form of the quadratic equation called vertex form.!!(ℎ,8) is the vertex of the graph.
