How is the graph of #h(x)=-x^2-2# related to the graph of #f(x)=x^2#?

Question

Avery Alexander · Accepted Answer

We are given

#y = f(x) = x^2 and# ... Equation.1

#y = h(x) = -x^2-2 and# ... Equation.2

We need to explain how these graphs are related.

Please read the explanation.

Graph of the Parabola for the Quadratic Equation..

#color(red)(y = f(x)= x^2)# ... Equation.1

..opens up as the coefficient of #color(red)(x^2)# term is greater than ZERO.

Vertex is at #(0,0)#

We have a minimum value for this parabola.

The Vertex is on the Line of Symmetry of the parabola.

Line of Symmetry at x = 0 is the imaginary line where we could fold the image of the parabola and both halves match exactly.

In our problem, this is also the y-axis

Graph #color(red)(y = x^2)# is available below:

Next, we will consider the equation #color(red)(y=f(x)=-x^2#

For this equation the parabola opens down as the coefficient of #color(red)(x^2)# term is less than ZERO.

Vertex is at #(0,0)#

Line of Symmetry at x = 0

We have a maximum value for this parabola.

This parabola is a reflection of the graph of #color(red)(y = x^2#

Graph #color(red)(y = -x^2)# is available below:

Next we will consider the equation

#color(red)(y = h(x) = -x^2-2 and# ... Equation.2

Vertex is at #(0,-2)#

Line of Symmetry at x = 0

This parabola is a shift of the graph of #color(red)(y = x^2# by #2# units down.

Graph #color(red)(y f(x) = = -x^2 - 2)# is available below:

Jameson Allen · Answer

undefined The graph of $h(x) = -x^2 - 2$ is the reflection of the graph of $f(x) = x^2$ over the x-axis and shifted downward by 2 units.