Solving Quadratics by Graphing

For the function below, find the direction of opening, the equation for the axis of symmetry, and the y-intercept. Use this information to sketch the graph.

y = -2x2 + 8x – 3

How could we find the vertex?

Warm Up

By hand: Find the AOS, then plug in to find the y-value of the vertex.

In the calculator: Graph the function, then press 2nd TRACE 3 (min.) or 4 (max.)

We can find the vertex by hand or in the calculator

y = -2x2 + 8x – 3Example:

y = 3x2 – 12x + 11

y = -x2 – 4x – 1

Examples:

Homework Solutions

Work with your partners to complete the investigation at the top of your notes.

Investigation

When a>1, the graph stretches (gets narrower)

When 0<a<1, the graph compresses (gets wider)

Summarize

Complete p. 257 (1-4) in your workbook

Apply your knowledge

1.C2.G3.D4.F

Solutions:

What does it mean to solve the equation 4x – 1 = 7? How would you solve it?

How might we solve the equation

x2 – 2x = 15?

SOLVING QUADRATICS

Our goal is to find the x-values that make an equation true. This was simple for linear equations, but quadratics require a little more effort.

Solving quadratics

GraphingFactoring (yay!)Completing the squareQuadratic formula

We will spend about a day on each of these!

Four ways to solve quadratics:

How are the following terms related?SolutionRootZerox-intercept

How many do we expect for quadratics?

Solving by graphing

What is another word for x-intercept?

What are the x-intercepts of the quadratic function graphed below?

What are the roots of this function?

Sometimes it is difficult to determine the exact solutions to a quadratic equation. Fortunately, our calculators can help.

What are the zeros of this quadratic?

To find the solutions/roots/x-intercepts/zeros:Step 1: Set Equation = 0 Step 2: Let Y1 = 0 Step 3: Let Y2 be the quadratic portion of

the equation.Step 4: Press GraphStep 5: Press Trace to one of the x –

interceptsStep 6: Press 2nd calc, 5, Enter, Enter, Enter.Step 7: Repeat 5 – 6 for other intercept

Solving by graphing

x2 – 2x = 15

x2 + 12 = 0

Examples:

14133 2 xx

7x2 + 4x – 9 = 0

z2 + 7z = -10

x2 = 10x – 24

You try!

Practice

Quiz