Quadratics Day 1

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Identifying Quadratic Functions

Test Reflection Questions

•

_{How do you feel about your score?}

•

_{How did you prepare for the test?}

•

_{What can you do to prepare for the}

next test?

•

_{How do you want to perform on the}

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Identifying Quadratic Functions

-Identify quadratic functions and determine whether they have a minimum or maximum.

-Graph a quadratic function and give its domain and range.

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Identifying Quadratic Functions

quadratic function

parabola

vertex

minimum

maximum

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Identifying Quadratic Functions

The function y = x2 is shown in the graph.

Notice that the graph is not linear. This function is a quadratic function. A quadratic function is any function that can be written in the standard form y = ax2 + bx + c, where a, b, and c are

real numbers and a ≠ 0. The function y = x2 can

be written as y = 1x2 + 0x + 0, where a = 1,

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Identifying Quadratic Functions

Since you are given an equation, use y = ax2 + bx + c.

Teacher Example 1: Identifying Quadratic Functions

Tell whether the function is quadratic. Explain.

y = 7x + 3

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Identifying Quadratic Functions

Tell whether the function is quadratic. Explain.

This is a quadratic function because it can be written in the form y = ax2 + bx + c where a = 2, b = _–1, and c

= 0.

y + x = 2x2

Try to write the function in the form y = ax2 + bx + c by

solving for y. Subtract x from both sides.

– x _– x y + x = 2x2

y = 2x2 _– x

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Identifying Quadratic Functions

Student Example 2

Tell whether the function is quadratic. Explain.

This is a quadratic function because it can be written in the form y = ax2 + bx + c where a = 10, b = 0, and c

=9.

y – 10x2 = 9

Try to write the function in the form y = ax2 + bx + c by

solving for y. Add 10x2 to

both sides.

+ 10x2 +10x2 y – 10x2 = 9

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Identifying Quadratic Functions

The differences between y-values for a constant

change in x-values are called first differences. It has constant second differences. This is true for all

quadratic functions.

Be sure there is a constant change in x -values before you try to find first or second differences.

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Identifying Quadratic Functions

Teacher Example 1: Identifying Quadratic Functions Tell whether the function is quadratic. Explain.

Since you are given a table of ordered pairs with a constant change in x-values, see if the

second differences are constant.

Find the first differences, then find the second differences.

The function is not quadratic. The second differences are not constant.

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Identifying Quadratic Functions

Student Example 1

Tell whether the function is quadratic. Explain.

Since you are given a table of ordered pairs with a constant change in

x-values, see if the second differences are constant.

Find the first differences, then find the second differences.

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Identifying Quadratic Functions

The graph of a quadratic function is a curve called a parabola. To graph a quadratic function,

generate enough ordered pairs to see the shape of the parabola.

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Identifying Quadratic Functions

Teacher Example: Graphing Quadratic Functions by Using a Table of Values

Use a table of values to graph the quadratic function.

y = –4x2

x –2 –1 0 1 2 y 0 –4 –16 –4 –16

Make a table of values. Choose values of x and use them to find values of y.

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Identifying Quadratic Functions

Use a table of values to graph each quadratic function.

Student Example 1

y = x2 + 2

x –2 –1 0 1 2 y 2 3 3 6 6

Make a table of values. Choose values of x and use them to find values of y.

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Identifying Quadratic Functions

Use a table of values to graph the quadratic function.

Student Example 2

y = –3x2 + 1

x –2 –1 0 1 2 y 1 –2 –11 –2 –11

Make a table of values. Choose values of x and use them to find values of y.

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Identifying Quadratic Functions

CFU

shape of the graphs in relation to their equations?

• _{What happens to the}

graph when a is positive?

• _{What happens when a is}

negative?

• A parabola opens upward when

a > 0

• A parabola opens downward

when a < 0. y = –4x2 y = x2 + 2

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Identifying Quadratic Functions

Tell whether the graph of the quadratic

function opens upward or downward. Explain.

Teacher Example: y = 5x – 3x2

Student Example 1: f(x) = –4x2 – x + 1

**Remember you must put the equation in Standard Form**

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Identifying Quadratic Functions

The highest or lowest point on a parabola is the

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Identifying Quadratic Functions

Teacher Example: Identifying the Vertex and the Minimum or Maximum

Identify the vertex of each parabola. Then give the minimum or maximum value of the function.

The vertex is (–3, 2), and the minimum is 2.

The vertex is (2, 5), and the maximum is 5.

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Identifying Quadratic Functions

Student Example 1

Identify the vertex of each parabola. Then give the minimum or maximum value of the function.

The vertex is (3, –1), and the minimum is –1.

The vertex is (–2, 5) and the maximum is 5.

DOMAIN AND RANGE

The domain of a function is the set of x-values that make that

function true. The range of a

function is the set of y-values that make that

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Identifying Quadratic Functions

Teacher Example: Domain and Range Find the domain and range.

Step 1 The graph opens downward, so identify the maximum.

The vertex is (–5, –3), so the maximum is –3.

Step 2 Find the domain and range.

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Identifying Quadratic Functions

Student Example 1

Find the domain and range.

Step 1 The graph opens upward, so identify the minimum.

The vertex is (–2, –4), so the minimum is –4.

Step 2 Find the domain and range.

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Identifying Quadratic Functions

Student Example 2

Find the domain and range.

Step 1 The graph opens downward, so identify the maximum.

The vertex is (2, 3), so the maximum is 3.

Step 2 Find the domain and range.

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Identifying Quadratic Functions

Exit Ticket

Use the graph for Problems 3-5. 2. Identify the vertex.

3. Does the function have a

minimum or maximum? What is it?

4. Find the domain and range. D: all real numbers;

R: y ≤ –4

max; –4

(5, –4)

1. Is y = –x – 1 quadratic?

Explain. _{No; there is no x}2-term,

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Identifying Quadratic Functions

The value of c in a quadratic function determines not only the

value of the y-intercept but also a vertical translation of the graph of the function up or down the y-axis.

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Identifying Quadratic Functions

The value of a in a quadratic function determines the direction a parabola opens and the width of the parabola.