Lesson 12

Graphing the Standard Form (Part 1)

  • Let’s see how the numbers in expressions like \(\text-3x^2+4\) affect their graph.

Problem 1

Here are four graphs. Match each graph with a quadratic equation that it represents.

Graph A

A curve in an x y plane, origin O.

Graph B

A curve in an x y plane, origin O.

Graph C

A curve in an x y plane, origin O.

Graph D

A curve in an x y plane, origin O.
A:

Graph A

B:

Graph B

C:

Graph C

D:

Graph D

1:

\(y = x^2\)

2:

\(y = x^2 + 5\)

3:

\(y = x^2 +7\)

4:

\(y = x^2 - 3\)

Problem 2

The two equations \(y=(x+2)(x+3)\) and \(y=x^2 + 5x + 6\) are equivalent.

  1. Which equation helps find the \(x\)-intercepts most efficiently?
  2. Which equation helps find the \(y\)-intercept most efficiently?

Problem 3

Here is a graph that represents \(y = x^2\).

On the same coordinate plane, sketch and label the graph that represents each equation:

  1. \(y = x^2 -4\)
  2. \(y = \text-x^2 + 5\)
A curve in an x y plane, origin O.

Problem 4

Select all equations whose graphs have a \(y\)-intercept with a positive \(y\)-coordinate.

A:

\(y=x^2 + 3x - 2\)

B:

\(y=x^2 - 10x\)

C:

\(y=(x-1)^2\)

D:

\(y=5x^2-3x-5\)

E:

\(y=(x+1)(x+2)\)

Problem 5

  1. Describe how the graph of \(A(x)=|x|\) has to be shifted to match the given graph.
  2. Write an equation for the function represented by the graph.
Absolute value function with vertex at 2 comma 3.
(From Unit 4, Lesson 14.)

Problem 6

Here is a graph of the function \(g\) given by \(g(x) = a \boldcdot b^x\).

What can you say about the value of \(b\)? Explain how you know. 

Graph of a decreasing exponential function g, x y plane, origin O.
(From Unit 5, Lesson 13.)

Problem 7

  1. What are the \(x\)-intercepts of the graph that represents \(y = (x+1)(x+5)\)? Explain how you know.
  2. What is the \(x\)-coordinate of the vertex of the graph that represents \(y = (x+1)(x+5)\)? Explain how you know.
  3. Find the \(y\)-coordinate of the vertex. Show your reasoning.
  4. Sketch a graph of \(y = (x+1)(x+5)\).
(From Unit 6, Lesson 11.)

Problem 8

Determine the \(x\)-intercepts, the vertex, and the \(y\)-intercept of the graph of each equation.

equation \(x\)-intercepts vertex \(y\)-intercept
\(y=(x-5)(x-3)\)      
\(y=2x(8-x)\)      
(From Unit 6, Lesson 11.)

Problem 9

Equal amounts of money were invested in stock A and stock B. In the first year, stock A increased in value by 20%, and stock B decreased by 20%. In the second year, stock A decreased in value by 20%, and stock B increased by 20%.

Was one stock a better investment than the other? Explain your reasoning.

(From Unit 5, Lesson 15.)