Question

translate each graph as specified below. (a) the graph of $y = x^2$ is shown. translate it to get the graph of $y = x^2 - 5$.

Explanation:

Step1: Recall Vertical Translation Rule

For a function \( y = f(x) \), the transformation \( y = f(x) + k \) (or \( y = f(x)-|k| \) when \( k<0 \)) translates the graph vertically. If \( k>0 \), it's up; if \( k<0 \), it's down. Here, \( f(x)=x^2 \), and the new function is \( y = x^2 - 5 \), so \( k=-5 \), meaning a vertical translation down by 5 units.

Step2: Apply to Vertex

The vertex of \( y = x^2 \) is at \( (0,0) \). After translating down 5 units, the new vertex is at \( (0,0 - 5)=(0,-5) \). The shape of the parabola (opening, width) remains the same, only the position of the vertex (and thus the entire graph) changes. So we move each point on \( y = x^2 \) down 5 units, or use the vertex to guide the translation.

Answer:

To get the graph of \( y = x^2 - 5 \) from \( y = x^2 \), translate the graph of \( y = x^2 \) 5 units downward (each point on \( y = x^2 \), including the vertex \((0,0)\), moves 5 units down to \((0, - 5)\) and the parabola retains its shape).