Chapter 4 Quadratic Functions and Factoring Chapter 4 Pre-Requisite Skills page 234

4.1 Graph Quadratic Functions in Standard Form

Terms Quadratic Function: Can be written in standard form: ax² + bx + c (a ≠ 0 or it would be linear) Parabola: Shape of all quadratic function graphs.

Parent Function for Quadratics xy table for y = x² Graph for y = x² Identify vertex and axis of symmetry

EXAMPLE 1 Graph a function of the form y = ax 2 Find the axis of symmetry and vertex, then graph y = 2x 2. Compare the graph with the graph of y = x 2.

Axis of Symmetry and Vertex Can be found using the formula: x = -b/2a and will always be a vertical line. To find the vertex point, plug this value back into the equation and get the (x, y) coordinate

Graphing ax² + bx + c without using xy tables Steps 1.Find the axis of symmetry (x = -b/2a) 2.Plug result back in and find y, this gives you the vertex. 3.If it’s not already part of your vertex, use c: it’s the point where the graph crosses y. 4.If it is part of your vertex, plug anything in for x and get a new point. 5.Use symmetry to get point on the other side and draw parabola

EXAMPLE 2 Graph a function of the form y = ax 2 + c Graph y = – Compare the graph with the x graph of y = x 2

GUIDED PRACTICE for Examples 1 and 2 Graph the function. Compare the graph with the graph of y = x y = – 4x 2

GUIDED PRACTICE for Examples 1 and 2 2. y = – x 2 – 5

GUIDED PRACTICE for Examples 1 and 2 3. f(x) = x

Observations ax² + bx + c What does having a negative a do to the graph? An |a| > 1? |a| < 1? Only knowing the above equation, where will the graph cross the y-axis?

EXAMPLE 3 Graph a function of the form y = ax 2 + bx + c Graph y = 2x 2 – 8x + 6.

GUIDED PRACTICE for Example 3 Graph the function. Label the vertex and axis of symmetry. 4. y = x 2 – 2x – 1

GUIDED PRACTICE for Example 3 5. y = 2x 2 + 6x + 3

GUIDED PRACTICE for Example 3 6. f (x) = x 2 – 5x –

Maximum and Minimum Just like it sounds See bottom of page 238 If a is positive you will have a max If a is negative you will have a min. The y value is used to describe these What relationship do the max and mins have with the vertex?

EXAMPLE 4 Find the minimum or maximum value Tell whether the function y = 3x 2 – 18x + 20 has a minimum value or a maximum value. Then find the minimum or maximum value.

GUIDED PRACTICE for Examples 4 and 5 7. Find the minimum value of y = 4x x – 3.

FOIL (x – 3)(x + 2) (2x + 2)(x – 1)

EXAMPLE 5 Solve a multi-step problem Go - Carts A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for every $1 reduction in the price per racer. How can the owner of the go-cart track maximize weekly revenue ?

Example A video store sells about 150 DVDs a week at a price of $20 each. The owner estimates that for each $1 decrease in price, about 25 more DVDs will be sold each week. How can the owner maximize weekly revenue?

Homework 6 – 38 ev, 44 – 48, 55, 56, 58, 59