Question

graph the given functions, f and g, in the same rectangular coordinate system. describe how the graph of g is related to the graph of f

f(x) = -x²
g(x) = -x² + 2

use the graphing tool to graph the functions.

Explanation:

Step1: Identify base function f(x)

The base function is $f(x) = -x^2$, a downward-opening parabola with vertex at $(0,0)$.

Step2: Identify transformation for g(x)

Compare $g(x) = -x^2 + 2$ to $f(x)$. This is $g(x) = f(x) + 2$.

Step3: Determine transformation type

Adding a positive constant to a function shifts its graph vertically upward.

Step4: Graph key points for f(x)

For $f(x) = -x^2$:

• When $x=-2$, $f(-2) = -(-2)^2 = -4$
• When $x=0$, $f(0) = -(0)^2 = 0$
• When $x=2$, $f(2) = -(2)^2 = -4$

Step5: Graph key points for g(x)

For $g(x) = -x^2 + 2$:

• When $x=-2$, $g(-2) = -(-2)^2 + 2 = -2$
• When $x=0$, $g(0) = -(0)^2 + 2 = 2$
• When $x=2$, $g(2) = -(2)^2 + 2 = -2$

Answer:

The graph of $g(x)$ is the graph of $f(x)$ shifted 2 units vertically upward.

• The graph of $f(x) = -x^2$ is a downward-opening parabola with vertex at $(0, 0)$.
• The graph of $g(x) = -x^2 + 2$ is a downward-opening parabola with vertex at $(0, 2)$.