Quadratic Functions & Equations Chapter 4

4-1 Quadratic Functions & Transformations A parabola is the graph of a QUADRATIC FUNCTION, which you can write in the form f (x) = ax 2 + bx + c, where a ≠ 0. Essential Understanding – The graph of any quadratic function is a transformation of the graph of the PARENT function y = x 2.

Quadratic Functions ▪The vertex form of a quadratic function is f (x) = a(x – h) 2 + k, where a ≠ 0. ▪The axis of symmetry is a line that divides the parabola into two mirror images. The equation of the axis of symmetry is x = h. The vertex of the parabola is (h, k), the intersection of the parabola and its axis of symmetry.

Problem 1 Graphing a Function of the form f(x) = ax 2 ▪ What is the graph of

Problem 1 – Got it? Graphing a Function of the form f(x) = ax 2 ▪ What is the graph of Reasoning What can you say about the graph of the function if a is a negative number? Explain.

Problem 2 Graphing a Function of the Form Graph each function. How is each graph a translation of?

Problem 2 – Got it? Graphing a Function of the Form Graph each function. How is each graph a translation of?

Vertex Form ▪ The vertex form, f (x) = a(x – h) 2 + k, where a ≠ 0 gives you info about the graph without drawing it. ▪ If a > 0, then you have a parabola that opens upward and your vertex is a minimum. ▪ If a < 0, then you have a parabola that opens downward and your vertex is a maximum.

Problem 3Interpreting Vertex Form For f (x) = 3(x – 4) 2 – 2, what are the vertex, axis of symmetry, if vertex is min/max, the domain and range? Vertex Axis of Symmetr y Max/MinDomainRange ( 4, -2) x = 4 a = 3 which is positive so graph opens up so we have a min ( - ∞, ∞ )( - 2, ∞ )

Problem 3 - Got it?Interpreting Vertex Form For f (x) = -2 (x + 1) 2 + 4, what are the vertex, axis of symmetry, if vertex is min/max, the domain and range? Vertex Axis of Symmetr y Max/MinDomainRange ( -1, 4) x = -1 a = -2 which is negative so graph opens down so we have a max ( - ∞, ∞ )( - ∞, 4 )

Summary so far… ▪ You can use the vertex form of a quadratic function, f (x) = a(x – h) 2 + k, to transform the graph of f (x) = x 2. ▪ Stretch or compress the graph of f (x) = x 2 vertically by the factor |a|. ▪ If a < 0, reflect the graph across the x-axis (flip vertically) ▪ Shift the graph h units horizontally (“+” means left, “-“ means right) ▪ Shift the graph k units vertically (“+” means up, “-“ means down)

Problem 4Using Vertex Form What is the graph of f (x) = -2(x – 1) look like? State the transformations and sketch the graph. HorizontalRight 1 VerticalUp 3 ReflectionYes Stretch Or Compress Stretch Vertex(1, 3) Axis of Symmetry x = 1

Problem 4 – ContinuedUsing Vertex Form What steps transform the graph of f (x) = x 2 to f (x) = -2(x – 1) 2 + 3?

Problem 4- Got it?Using Vertex Form What is the graph of f (x) = 2(x + 2) look like? State the transformations and sketch the graph. HorizontalLeft 2 VerticalDown 5 ReflectionNone Stretch Or Compress Stretch 2 Vertex(-2, -5) Axis of Symmetry x = -2

Problem 5Writing a Quad Function in Vertex Form The picture shows the jump of a dolphin. What quadratic function models the path of the dolphin’s jump?

4.9 Concept Byte – Quadratic Inequalities ▪ y > x 2 + 5x – 6 Steps for graphing a quadratic inequality: 1.Graph the parabola (using standard or vertex form…whichever form the problem is given.) 2.Remember solid vs. dotted lines 3.Shade accordingly:

4.9 Concept Byte – Quadratic Inequalities ▪ y > -(x +1) 2 + 8y < (x – 4) 2 – 5 ▪

4-2 Standard form of Quadratic Functions To graph a quadratic function written in standard form. Page 11

Standard form of a Quadratic ▪ In the previous section, you worked with quadratic functions in vertex form: f (x) = a(x – h) 2 + k ▪ Now you will use quadratic functions in standard form: f (x) = ax 2 + bx + c, where a ≠ 0. Essential Understanding ▪ For any quadratic function f (x) = ax 2 + bx + c, the values of a, b, and c provide key information about its graph. ▪ You can find information about the graph of a quadratic function (such as the vertex) easily from the vertex form. ▪ Such information is “hidden” in standard form. However, standard form is easier to graph using a calculator.

Problem 1 Finding Features of a Quadratic Equation What are the vertex, axis of symmetry, the max/min, and the range of y = 2x 2 + 8x – 2? Use your graphing calculator for help. xf 1(x) Vertex -58( -2, -10) -4-2 Max/Min -3-8Min Axis of Symmetry? -8x = Range? 18 ( -10, ∞ )

Problem 1 – Got it? Finding Features of a Quadratic Equation What are the vertex, axis of symmetry, the max/min, and the range of y = -3x 2 - 4x + 6? Use your graphing calculator for help. xf 1(x) Vertex -5-49(-1, 7) Max/Min -3-9Max -22 Axis of Symmetry? 7X = Range? 1( -∞, 7)

Key Concept How do you Graph when you don’t have a calculator? ▪ In order to graph when you don’t have a calculator, you only need to find a few specific points in order to “fake” the graph. Read the properties below. ▪ The graph of, f (x) = ax 2 + bx + c, where a ≠ 0 is a parabola. ▪ It has a “U” shape. ▪ If a > 0, then you have a parabola that opens up. If a < 0, then you have a parabola that opens down. ▪ The axis of symmetry is the line x = -b / (2a) ▪ The x-coordinate of the vertex is -b / (2a) ▪ To find the y-coordinate of the vertex, plug in what you got when you found the x-coordinate. ▪ The y-intercept is ( 0, c).

Problem 2 Graphing a Function of the form y = ax 2 + bx + c What is the graph of y = x 2 + 2x + 3 Step 1Identify a, b, and c. Step 2Find the AXIS OF SYMMETRY, then sketch it. Step 3Find the x-coordinate of the vertex, then the y. (, ) Step 4Find the y-intercept. (, ) Step 5Determine which way the parabola opens. Step 6Connect everything with a smooth “U” shaped curve.

Problem 2 – Got it? Graphing a Function of the form y = ax 2 + bx + c What is the graph of y = -2x 2 + 2x - 5 Step 1Identify a, b, and c. Step 2Find the AXIS OF SYMMETRY, then sketch it. Step 3Find the x-coordinate of the vertex, then the y. (, ) Step 4Find the y-intercept. (, ) Step 5Determine which way the parabola opens. Step 6Connect everything with a smooth “U” shaped curve.

Problem 3 Converting Standard Form to Vertex Form What is the vertex form of y = 2x x + 7 Step 1Identify a and b from the equation Step 2Find the x-coordinate of the vertex. Step 3Use your value of the x-coordinate to solve for the y- coordinate of the vertex. Step 4Write the vertex as (h, k). Step 5Substitute the values of a, h, and k into the vertex form y = a (x – h) 2 + k.

Problem 3 - Got it? Converting Standard Form to Vertex Form What is the vertex form of y = -x 2 + 4x - 5 Step 1Identify a and b from the equation Step 2Find the x-coordinate of the vertex. Step 3Use your value of the x-coordinate to solve for the y- coordinate of the vertex. Step 4Write the vertex as (h, k). Step 5Substitute the values of a, h, and k into the vertex form y = a (x – h) 2 + k.

4-3 Modeling with Quadratic Functions To model data with quadratic functions. Page 15

Think about it… You and a friend are tossing a ball back and forth. You each toss and catch the ball at waist level, about 3 ft high. What type of equation models the path of the ball?

▪ When you know the vertex and a point on a parabola, you can use vertex form, f (x) = a(x – h) 2 + k, to write an equation of the parabola. ▪ If you do not know the vertex, you can use standard form : f (x) = ax 2 + bx + c, and any two points of the parabola to find an equation. Essential Understanding: Three non-collinear points, no two of which are in line vertically, are on the graph of exactly one quadratic function.

Problem 2Comparing Quadratic Models PhysicsCampers at an aerospace camp launch rockets on the last day of camp. The path of Rocket 1 is modeled by the equation h = -16t t + 1, where t is time in seconds and h is the distance (height) off the ground. The path of Rocket 2 is modeled by the graph. Which rocket flew higher? (a) Find the max height of each rocket. (b) Which rocket stayed in the air longer? (c) What is the reasonable domain? What is the reasonable range?

When more than three data points suggest a quadratic function, you can use the quadratic regression feature of a graphing calculator to find a quadratic model. Problem 3Using Quadratic Regression The table shows a meteorologist’s predicted temperatures for an October day in Sacramento, CA. a)What is a quadratic model for this data? b)Use your model to predict the high temperature for the day. At what time does that happen?

Steps to follow… 1.CTRL + then #4:ClrAllLists 2.Clear any data in Y= and make sure PLOTS are turned OFF 3.STAT #1:Edit (enter data into L1 & L2…using L1 for “x” and L2 for “y”) 4.Copy equation onto your paper following ax 2 + bx + c (calculator will give you values for a,b,c…round to 2 decimal places) 5.Type your function into Y= (Y1) and GRAPH the function. 6.You can use various tools now to answer questions… 2 nd – CALC #2: zeros (xintercepts – find one at a time using this option) #3: minimum (vertex that is a minimum when parabola opens up) #4: maximum (vertex that is a maximum when parabola opens down) 2 nd – GRAPH (table option to see values of x and corresponding y) (Don’t forget you can use too!)

Problem 3Got it?Using Quadratic Regression The table shows a meteorologist’s predicted temperatures for an summer day in Denver, CO. a)What is a quadratic model for this data? b)Use your model to predict the low temperature for the day. At what time does that happen?