Chapter 1.2, Problem 42E

Solution

a.

To predict: The changes in the graph of the function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ , when $a$ changes to $-3$ .

The revised function $y=-3{(x-3)}^2-2$ opens in oppositive direction as that of the original function and is compressed in y-direction by a factor of 3.

Given information:

The function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ is

The value $a$ changes to $-3$ .

Concept used:

For a function of the type $y=\text{C}\cdot f\left(x\right)$ , if $\text{C}>1$ , it compresses the function in the x-direction and if $0<\text{C}<1$ , it stretches the function in x-direction.

Calculation:

The function $y={(x-3)}^2-2$ is a quadratic function with $a=1$ . This represents an upward parabola.

Multiply the term ${(x-3)}^2$ with negative. Then, the function becomes, $y=-{(x-3)}^2-2$ . This shows that the revised function is a downward parabola.

Now, multiply the term $-{(x-3)}^2$ by 3. So, the function becomes $-3{(x-3)}^2$ .

Since $3>1$ , the function $y=-3{(x-3)}^2-2$ is compressed in y-direction.

Therefore, the function $y=-{(x-3)}^2-2$ is downward parabola and is compressed in y-direction to get the graph of the function $y=-3{(x-3)}^2-2$ .

Now, graph the original function and the revised function using a graphing calculator.

  

The graph shows that the prediction about the revised function is correct.

Conclusion:

The revised function $y=-3{(x-3)}^2-2$ opens in oppositive direction as that of the original function and is compressed in y-direction by a factor of 3.

b.

To predict: The changes in the graph of the function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ , when $h$ changes to $-1$ .

The revised function is shifted 4 units to the left of that of the original function.

Given information:

The function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ is

The value $h$ changes to $-1$ .

Concept used:

For a function of the type $y=f\left(x+C\right)$ , if $C>0$ , the original function $f\left(x\right)$ shifts $C$ units to the left and if $C<0$ , the original function $f\left(x\right)$ shifts $C$ units to the right.

Calculation:

The function $y={(x-3)}^2-2$ has vertex at the point $\left(3,-2\right)$ .

When $h$ changes to $-1$ , the function becomes $y={(x+1)}^2-2$ . This has the vertex at the point $\left(-1,-2\right)$ .

That means, the revised function is shifted 4 units to the left of that of the original function.

Now, graph the original function and the revised function using a graphing calculator.

  

The graph shows that the revised function is shifted 4 units left to that of the original function.

Conclusion:

The revised function is shifted 4 units to the left of that of the original function.

c.

To predict: The changes in the graph of the function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ , when $k$ changes to $2$ .

The revised function is shifted 4 units to the left of that of the original function.

Given information:

The function $y=a{(x-h)}^2+k$ , where $a=1$ , $h=3$ and $k=-2$ is

The value $k$ changes to $2$ .

Concept used:

For a function of the type $y=f\left(x+C\right)$ , if $C>0$ , the original function $f\left(x\right)$ shifts $C$ units upward and if $C<0$ , the original function $f\left(x\right)$ shifts $C$ units downward.

Calculation:

The function $y={(x-3)}^2-2$ has vertex at the point $\left(3,-2\right)$ .

When $k$ changes to $2$ , the function becomes $y={(x-3)}^2+2$ . This has the vertex at the point $\left(3,2\right)$ .

That means, the revised function is shifted 4 units upwards to that of the original function.

Now, graph the original function and the revised function using a graphing calculator.

  

The graph shows that the revised function is shifted 4 units upwards to that of the original function.

Conclusion:

The revised function is shifted 4 units upwards to that of the original function.