May 16, 2016

Aim: To review for Quadratic Function Exam #2

Homework: Study Review Materials

Warm Up - Solve using factoring:

5x2 + 7x + 2 = 0

May 16, 2016

Review Topic Index

1. Consecutive Integer Word Problems

2. Pythagorean Theorem Word Problems

3. Finding zeroes/roots on calculator

4. Discriminant

5. Completing the Square

6. Writing Equation from Roots

7. Linear Quadratic Systems

8. Graphing Linear Quadratic System Word Problem

May 16, 2016

Concept Review: (1) Consecutive Integer Word Problems

Consecutive integers are 1 apart.

x

x + 1

x + 2

Consecutive odd or even integers are 2 apart.

x

x + 2

x + 4

Product means to multiply together.

Example

If there are THREE consecutive odd integers, the product of the two SMALLEST consecutive odd integers can be written as:

_________________________

May 16, 2016

Sample Question: (1) Consecutive Integer Word Problems

1

A

B

C

D

The ages of three babies are consecutive even integers. If the product of the two oldest babies is 100, which can be used to find their ages?

x2 + 4x - 100 = 0

(x + 2) + (x + 4) = 100

x2 + 3x - 98 = 0

2x + 6 = 100

May 16, 2016

Concept Review: (2) Pythagorean Theorem Word Problems

Pythagorean Theorem: In a right triangle, a2 + b2 = c2

c must be hypotenuse (opposite right angle)

Strategy:

Draw a right triangle. Label one side x.

Label other sides in terms of x.

Substitute into Pythagorean Theorem and solve using quadratic strategies.

Ex: In a right triangle, the hypotenuse is 2 longer than one leg and 7 longer than the other. Find the side lengths.

xx-2

x-7

May 16, 2016

Sample Question: (2) Pythagorean Theorem Word Problems

In a right triangle, one leg is 4 longer than the other. And the hypotenuse is 8 longer than the shorter leg. Find the dimensions of the right triangle algebraically.

May 16, 2016

Concept Review: (3) Finding Roots on a Calculator

To find roots on a calculator, use:

2nd - Calc - Zeros

Left and Right Bound should be placed correctly. Guess should be placed anywhere near the root.

If you can't see the graph nicely:

Zoom - Standard or

Zoom - Fit or

Zoom - Out or

Left Bound

Left Bound

Right Bound

Right Bound

May 16, 2016

Sample Question: (1) Finding Roots on a Calculator

2

A

B

C

D

The zeros of the function g(x) = 4x2 + 16x - 84

7 and -3

-2 and 100

-7 and 3

2 and -100

May 16, 2016

Sample Question: (2) Finding Roots on a Calculator

3

A

B

C

D

The zeros of the function g(x) = x3 - 25x

-5

-5 and 0

-5 and 5

-5, 0, and 5

May 16, 2016

Concept Review: (4) The Discriminant

The discriminant is found in the quadratic formula

It describes how many roots there are and tells you about the roots.

If perfect square, roots are rational. If not, roots are irrational.

b2 - 4ac < 0

no real rootsb2 - 4ac = 0

1 real root

b2 - 4ac > 0

no real roots

May 16, 2016

Sample Question: (4) Discriminant

Calculate the discriminant for the function f(x) = -2x2 + 5x - 3

How many real roots does f(x) have?

May 16, 2016

Calculate the discriminant for the function g(x) = 11x2 + 10x - 1

How many real roots does f(x) have? Are they rational or irrational?

How could you check your solution using your calculator?

Sample Question: (4) Discriminant

May 16, 2016

Concept Review: (5) Completing the Square

Steps for CTS:

1. Make a = 1 (Divide every term by "a")

2. Move constant (c) over

3. Add (b/2)2 to both sides to create perfect binomial

4. Move constant/values back over to make into "f(x)" form

Standard Form: f(x) = Ax2 + Bx + C

Vertex Form: f(x) = a(x - h)2 + k

Be Careful: Don't forget about the value you divided out in step 1. Since this isn't an equation equal to 0, but rather a function, it does not simply go away and must be preserved.

Helpful hint: When in vertex form, the sign of the number in the parentheses will match the sign of "B" in standard form.

May 16, 2016

Sample Question: (5) Completing the Square

Solve the following quadratic using completing the square.

x2 + 8x - 2 = 0

May 16, 2016

Solve the following quadratic using completing the square.

x2 + x - 5 = 0

Sample Question: (5) Completing the Square

May 16, 2016

Sample Question: (5) Completing the Square

Which of the following could be one of the steps to solve the equation x2 - 5x + 1 = 0

a) (x + 5/2)2 = 21/4

b) (x + 5)2 = 21/4

c) (x - 5)2 = 21/4

d) (x - 5/2)2 = 21/4

May 16, 2016

Concept Review: (6) Writing the Equation from Roots

Let's make a function with roots a and b.

Roots make the output, f(x), equal to zero. So, we want to design an equation so that f(x) becomes zero when we replace x with a or b.

Which one will work?

f(x) = (x - a)(x - b) OR f(x) = (x + a)(x + b)

a b

May 16, 2016

Sample Question: (1) Writing an Equation from Roots

4 Create a function with roots -4 and -3. Be sure to expand any binomials.

May 16, 2016

Sample Question: (2) Writing an Equation from Roots

5 Write a function with roots equal to 8, 0, and -8. Be sure to expand any product of binomials.

May 16, 2016

Concept Review: (7) Quad-Linear System of Equations

Quad: y = ax2 + bx + c

Linear y = mx + b

The solutions to the system are the points that make both equations true. So, where the parabola and line intersect.

To solve algebraically:• Get y alone in both equations• Set equal to each other• Move all terms to one side (set equal to 0)• Solve using any method of solving quadratics.

Could have 0, 1, or 2 solutions.

May 16, 2016

Sample Question: (7) Quad-Linear Systems

Ex 1: Find the solutions to the system below algebraically

f(x) = 3x2 + 4x - 7

g(x) = 2x + 1

May 16, 2016

Sample Question: (7) Quad-Linear Systems

f(x) = 2x2 - 4x - 7

g(x) = 5x - 2

Bob is solving a system of equations and arrives at the step (2x + 1)(x - 5) = 0. Which of the following could represent the original system?

a) b) f(x) = 2x2 + 4x - 9

g(x) = 3x - 4

May 16, 2016

Sample Question: (7) Quad-Linear Systems

Ex 1: Find the solutions to the system below algebraically

f(x) = x2 + 8x - 12

g(x) = - 2x + 12

May 16, 2016

Concept Review: (8) Graphing Linear Quadratic System Word Problems

Time

Out

put (

Tem

pera

ture

, $, e

tc.) The ___ - coordinate of this point tells

how long it took for the output of one graph to equal the output of the other.

Steps: Solve the system. Find the ___ - coordinate.

This distance tells how much higher the output of one graph is compared to the other.

Steps: Find the outputs. Subtract them.

May 16, 2016

Sample Question: (1) Graphing Linear Quadratic System Word Problems

6 The height, h, of a baseball in meters at t seconds after it is tossed out of a window is modeled by: h = -5t2 + 20t + 15. A bird flies toward the baseball. The trajectory of the bird is given by the equation h = 3t + 3. Will the bird hit the baseball? If so, when?

At 2 seconds, how much higher will the baseball be compared to the bird?

May 16, 2016