Graphing Quadratic Functions

Table of Contents Introduction to Graphing Quadratics (19.1) Graphing in Vertex Form Using Transformations (19.2) Graphing in Standard Form (19.3) Graphing in Factored Form (20.1 and 20.2)

Introduction to Graphing Quadratics

Quadratics Definition: Equations and expressions involving polynomials where the highest power is 2. Graphically: always a “U” shape Algebraically: Values increase at an increasing rate Also named parabola

Parts of Quadratic Graph The bottom (or top) of the U is called the vertex, or the turning point. The vertex of a parabola opening upward is also called the minimum point. The vertex of a parabola opening downward is also called the maximum point. The x-intercepts are called the roots, or the zeros. To find thex-intercepts, set ax2 + bx + c = 0. The ends of the graph continue to positive infinity (or negative infinity) unless the domain (the x's to be graphed) is otherwise specified.  

Vertex: Minimum vs. Maximum

Sketch the Graph

Sketch the Graph g(x)= - 3x2 Y -3 -2 -1 1 2 3 Domain: ____________ Range: _____________

Write a Function Given a Graph Use form g(x) = ax2 Plug in x-cord for x and y-cord of g(x) and solve for “a”

Write a Function Given a Graph

Satellite dishes reflect radio waves onto a collector by using a reflector dish shaped like a parabola. The graph shows the height h in feet of the reflector relative to the distance x in feet from the center of the satellite dish. Find the equation of the quadratic and describe what the function represents.

Equation:___________________ Describe: Vertex? Point: (60, 12)?

Graphing in Vertex Form Using Transformations

3 Forms of Quadratic Equations Vertex Form: y= a(x – h)2 + k Standard Form: y = ax2 + bx + c Factored Form: y = k(x – a)(x – b)

Graphing in Vertex Form Identify the vertex. Plot vertex and draw axis of symmetry. Create table of values (pick 2 x-values bigger than vertex and 2 x-values smaller than vertex.)

Graph: y = (x + 3)2 - 1 Identify the Vertex:_______ Is the vertex a min or max? End Behavior? X Y

Graph: y = -(x – 2)2 + 4 Identify the Vertex:_______ Is the vertex a min or max? End Behavior? X Y

Up or Down?!

Skinny or Fat?

Transformations of Vertex Form Type of Transformation Details a h k

Graph: y = 2(x + 3)2 - 1 Identify the: Vertex:________________ Is the vertex a min or max? Vertical Transformation? Horizontal Transformation?

Graph: y = 𝟏 𝟐 (x - 4)2 + 2 Identify the: Vertex:________________ Is the vertex a min or max? Vertical Transformation? Horizontal Transformation?

Graphing Using Standard Form

Standard Form of Quadratics y = ax2 + bx + c Conditions: a, b and c must be real numbers and not be zero.

Standard Form: Find Vertex Identify a, b and c. Find x-coordinate using formula: −𝑏 2𝑎 Plug in x-cord into original equation and solve for y-coordinate.

Let’s Practice… 1. Calculate the vertex. y = - x2 – 8x – 15 a:________ b: _________ c:_________ x-cord: y-cord:

Let’s Practice… 2. Calculate the vertex. y = x2 – 4x +5 a:________ b: _________ c:_________ x-cord: y-cord:

Find Zeros/Roots Factor original equation. Set factors equal to zero and solve. The two solutions are the x-intercepts.

Let’s Practice… 1. Find zeros/roots. y = x2 + 8x + 15 Factor: Solve: x-intercepts:

Graph: y = x2 + 5x + 4 Identify a:______ b:_____ c: _____ Vertex:________________ Zeros:

Graph: 6x + 8 = -x2 Identify a:______ b:_____ c: _____ Vertex:________________ Zeros:

The equation for motion of a projectile fired straight up at an initial velocity of 64 ft/s is ℎ=64𝑡− 16𝑡 2 , where h is height in feet and t is time in seconds. Find the time the projectile needs to reach its highest point. How high will it go?

Graph: 2x2 – 5 = - 3 Identify a:______ b:_____ c: _____ Vertex:________________ Zeros:

A baseball coach used a pitching machine to simulate pop flies during practice. The quadratic function h(t) = -16t2 + 80t + 5 models the height in feet of the baseball after t seconds. The ball leave the pitching machine and is caught at a height of 5 feet. How long is the baseball in the air?

Graph: 3x2 – 9 = -6 Identify a:______ b:_____ c: _____ Vertex:________________ Zeros:

Graphing in Factored Form

Graph: Factored Form Factored Form: y = k(x – a)(x – b) Set factors (parentheses) equal to zero and solve. Plot on graph. X-value of vertex is half way between zeros (from step 1). Plug in x-value to equation to y-value of vertex.

Graph: y = (x – 1)(x – 3) Identify the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

A tennis ball is tossed upward from a balcony A tennis ball is tossed upward from a balcony. The height of the ball in feet can be modeled by the function y = -4(2x + 1)(2x – 3) where x is the time in seconds after the ball is released. Find the maximum height of the ball and the time it takes the ball to reach this height. Determine how long it takes the ball to hit the ground.

Graph: y = 2(x + 4)(x + 2) Identify the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

Graph: y = x2 – 4x - 5 Identify the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

Graph: 6x + 8 = -x2 Identify the: Zeros: _________________ Vertex:________________ Is the vertex a min or max?

Mini Quiz