Question

which graph represents the parent function of $y = x^2 - 5$?
a graph of a downward - opening parabola with vertex at the origin
b graph of an upward - opening parabola with vertex on the y - axis above the origin
c graph of an upward - opening parabola with vertex on the x - axis to the right of the origin
d graph of an upward - opening parabola with vertex at the origin

Explanation:

Step1: Identify the parent function

The given function is \( y = x^2 - 5 \). The parent function of a quadratic function \( y = ax^2 + bx + c \) is \( y = x^2 \). So we need to find the graph of \( y = x^2 \).

Step2: Analyze the graph of \( y = x^2 \)

The graph of \( y = x^2 \) is a parabola opening upwards with vertex at the origin \((0,0)\). Let's check the options:

• Option A: The parabola opens downwards (vertex at \((0,0)\) but opens down), so it's \( y = -x^2 \), not the parent of \( y = x^2 - 5 \).
• Option B: The parabola has vertex not at the origin (it's shifted), so it's a transformed function, not the parent.
• Option C: The parabola is shifted horizontally, not the parent \( y = x^2 \).
• Option D: The parabola opens upwards with vertex at \((0,0)\), which matches the graph of \( y = x^2 \), the parent function of \( y = x^2 - 5 \).

Answer:

D (assuming D is the graph with vertex at (0,0) opening upwards, as per the analysis of \( y = x^2 \))