Chapter 9.3, Problem 34PPS

(a)

To determine

Graph the family of equations: $y=x^2;y={(x-2)}^2;y={(x-4)}^2;y={(x+3)}^2;y={(x+5)}^2$ on the same screen. Describe how the graph of the functions are related to the graph $f(x)=x^2$

Answer to Problem 34PPS

Explanation of Solution

Given:

The equations: $y=x^2;y={(x-2)}^2;y={(x-4)}^2;y={(x+3)}^2;y={(x+5)}^2$

Concept Used:

Rule 1: The graph of the equation $y={(x+h)}^2$ is shifted h unit left from the parent graph $y=x^2$

Rule 2: The graph of the equation $y={(x-h)}^2$ is shifted h unit right from the parent graph $y=x^2$

The graph of $y=x^2$ is the parent function and its vertex is (0, 0)

The graph of $y={(x-2)}^2$ is shifted 2 units right from the parent graph $y=x^2$

The graph of $y={(x-4)}^2$ is shifted 4 units right from the parent graph $y=x^2$

The graph of $y={(x+3)}^2$ is shifted 3 units left from the parent graph $y=x^2$

The graph of $y={(x+5)}^2$ is shifted 5 units right from the parent graph $y=x^2$

(b)

To determine

Write a concept for quadratic functions, similar to the concept for vertical translations, to describe the effect of a value being added to or subtracted from x inside the parenthesis.

Answer to Problem 34PPS

Explanation of Solution

Given:

The functions: $y=x^2;y={(x-2)}^2;y={(x-4)}^2;y={(x+3)}^2;y={(x+5)}^2$

Concept Used:

Horizontal Translation:

Rule 1: The graph of the equation $y={(x+h)}^2$ is shifted h unit left from the parent graph $y=x^2$

Rule 2: The graph of the equation $y={(x-h)}^2$ is shifted h unit right from the parent graph $y=x^2$

Vertical Translation:

Rule 1: The graph of the equation $y=x^2+c$ is shifted c unit up from the parent graph $y=x^2$

Rule 2: The graph of the equation $y=x^2-c$ is shifted c unit down from the parent graph $y=x^2$

(c)

To determine

Predict where the graphs of $y={(x-7)}^2$ and $y={(x+4)}^2$ will be located.

Answer to Problem 34PPS

Explanation of Solution

Given:

The functions: $y={(x-7)}^2;y={(x+4)}^2$

Concept Used:

Horizontal Translation:

Rule 1: The graph of the equation $y={(x+h)}^2$ is shifted h unit left from the parent graph $y=x^2$

Rule 2: The graph of the equation $y={(x-h)}^2$ is shifted h unit right from the parent graph $y=x^2$

The graph of $y={(x-7)}^2$ is shifted 7 units right from the parent graph $y=x^2$

The graph of $y={(x+4)}^2$ is shifted 4 units left from the parent graph $y=x^2$