Chapter 6.8, Problem 2STP

To determine

To Describe: The effect when the graph: $y=x^2+4$ is change to $y=x^2-3$ .

Answer to Problem 2STP

The graph had the same shape from the first expression to its second then vertex is shifted down: $D$option is right one.

Explanation of Solution

Given information:

The effect when the graph: $y=x^2+4$ is change to $y=x^2-3$ .

Then the options are:

F. The slope of the graph changes.

G. The graph widens.

H. The graph is the same shape, and the vertex of the graph is moved down.

J. The graph is the same shape, and the vertex of the graph is shifted to the left.

Formula used:

The formula for finding the slope:

  $m=\frac{y_2-y_1}{x_2-x_1}$

For finding the vertex:

The vertex point of the graph is the maxima and the minima point of the graph.

For finding the $y$ -coordinate if the $x$ coordinates are given:

  $\frac{-b}{2a}$

Consider the equation,

  $y=x^2+4$

First draw the graph for the equation $y=x^2+4$ ,as below:

  

Here this graph shows that in the equation $y=x^2+4$ there is increment of four points in the parabola upward above the $x$ -axis that means the graph shifted four points upward.

Now the graph for another equation $y=x^2-3$ is:

  

Here this graph shows that in the equation $y=x^2-3$ there is decrement of three points in the parabola upward below the $x$ -axis that means the graph shifted three points below the $x$ -axis.

Now when compare these two graphs, get the observation:

The difference between two graphs is not the slope.

These two graphs are in the slope intercepts form of $y=mx+b$ , where $m$ is the slope and $b$ is the $y$ -intercept .

The $y$ -intercept of a graph is also the vertex of the graph.

So, they have the same shape from the first expression to its second then vertex is shifted down.

Thus, the graph had the same shape from the first expression to its second then vertex is shifted down: $D$option is right one.