Question

8-44. match each rule below with its corresponding graph. can you do this without making any tables? explain your selections.
a. $y = -x^2 - 2$
b. $y = x^2 - 2$
c. $y = -x^2 + 2$
1.
graph 1: a downward-opening parabola with vertex at (0, 2), crossing the x-axis
2.
graph 2: a downward-opening parabola with vertex at (0, -2)
3.
graph 3: an upward-opening parabola with vertex at (0, -2)

Explanation:

1. For $y = -x^2 - 2$: The negative coefficient of $x^2$ makes the parabola open downward. The constant term $-2$ shifts the vertex of $y=-x^2$ down 2 units, matching Graph 2.
2. For $y = x^2 - 2$: The positive coefficient of $x^2$ makes the parabola open upward. The constant term $-2$ shifts the vertex of $y=x^2$ down 2 units, matching Graph 3.
3. For $y = -x^2 + 2$: The negative coefficient of $x^2$ makes the parabola open downward. The constant term $+2$ shifts the vertex of $y=-x^2$ up 2 units, matching Graph 1.

Answer:

a. $y = -x^2 - 2$ matches Graph 2
b. $y = x^2 - 2$ matches Graph 3
c. $y = -x^2 + 2$ matches Graph 1