Question

use your graphing calculator to sketch the graph of the following quadratic function, then determine the domain and range.

$y = -x^2 + 7$

sketch the graph of the function $y = -x^2 + 7$. choose the correct graph below.

\\(\bigcirc\\) a.
\\(\bigcirc\\) b.
\\(\bigcirc\\) c.
\\(\bigcirc\\) d.

-12, 12, 4 by -12, 12, 4

Explanation:

Step1: Analyze the quadratic function

The function is \( y = -x^2 + 7 \), which is a quadratic function in the form \( y = ax^2 + bx + c \) with \( a=-1 \), \( b = 0 \), \( c = 7 \). Since \( a=-1<0 \), the parabola opens downward.

Step2: Find the vertex

The vertex of a quadratic function \( y = ax^2+bx + c \) has its x - coordinate at \( x=-\frac{b}{2a} \). Here, \( b = 0 \) and \( a=-1 \), so \( x = 0 \). Substituting \( x = 0 \) into the function, we get \( y=-0^2 + 7=7 \). So the vertex is at \( (0,7) \).

Step3: Analyze the options

• Option A: The parabola opens upward (since it has a minimum at the vertex), which does not match \( a=-1<0 \) (should open downward).
• Option B: The vertex seems to be at a negative y - value, but our vertex is at \( y = 7 \), so this is incorrect.
• Option C: The parabola opens downward (matches \( a=-1<0 \)) and the vertex is at a positive y - value (consistent with \( y = 7 \) when \( x = 0 \)).
• Option D: The graph does not look like a parabola with the correct vertex and direction.

Answer:

C.