Chapter 9.3, Problem 35HP

(a)

To determine

Find the statements are sometimes, always or never true.

Answer to Problem 35HP

The graph of $y=x^2+c$ will have sometimes its vertex at the origin.

Explanation of Solution

Given:

The graph of $y=x^2+c$ has its vertex at the origin.

Concept Used:

The vertex of $y=x^2+c$ is $(0,c)$ . This will equal (0,0) only when $c=0$ . For any other value, the graph will be translated up or down.

Therefore, the graph of $y=x^2+c$ will have sometimes its vertex at the origin.

Thus, the graph of $y=x^2+c$ will have sometimes its vertex at the origin.

(b)

To determine

Find the statements are sometimes, always or never true.

Answer to Problem 35HP

The graph of $y=a{x}^2$ and its reflection $y=-a{x}^2$ over the x − axis will have always the same width.

Explanation of Solution

Given:

The graph of $y=a{x}^2$ and its reflection $y=-a{x}^2$ are the same width.

Concept Used:

The graph of $y=a{x}^2$ and its reflection $y=-a{x}^2$ , both graphs are dilated by a factor $a$ [ $\left|a\right|=\left|-a\right|$ ] and will have the same width.

Therefore, the graph of $y=a{x}^2$ and its reflection $y=-a{x}^2$ over the x − axis will have always the same width.

Calculation:

Thus, the graph of $y=a{x}^2$ and its reflection $y=-a{x}^2$ over the x − axis will have always the same width.

(c)

To determine

Find the statements are sometimes, always or never true.

Answer to Problem 35HP

The graph of $y=x^2+c$ and the graph of a quadratic with a vertex at $(0,5)$ will never have the same maximum or minimum point.

Explanation of Solution

Given:

The graph of $y=x^2+c$ opens downward

Concept Used:

The vertex of the graph of $y=x^2+c$ , if $c\ge 0$ is $(0,c)$ and it will be a minimum point since the graph will open up.

A quadratic with vertex of $(0,5)$ will have a minimum point at $(0,5)$ if the graph opens up and a maximum point at $(0,5)$ if the graph opens down.

Since $c\ge 0$ , the point cannot be the same.

Therefore, the graph of $y=x^2+c$ and the graph of a quadratic with a vertex at $(0,5)$ will never have the same maximum or minimum point.

Thus, the graph of $y=x^2+c$ and the graph of a quadratic with a vertex at $(0,5)$ will never have the same maximum or minimum point.