Question

which graph represents the function $f(x) = (x - 5)^2 + 3$? there are two graphs shown with radio buttons, but the specific details of the second graphs axes are partially visible with y-axis labeled and grid lines, x-axis not fully shown as in the first graph. the first graph has x-axis from -2 to 8 and y-axis from -2 to 8 with a parabola opening upwards, vertex around (5,3). the second graph has a y-axis from 1 to 8 and a parabola opening upwards with vertex at some x (not fully clear from ocr but structure similar).

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For \( f(x)=(x - 5)^2+3 \), \( h = 5 \) and \( k = 3 \), so the vertex is \((5, 3)\).

Step2: Analyze the direction of opening

Since the coefficient of \((x - 5)^2\) is \( 1 \) (positive), the parabola opens upwards.

Step3: Match with the graphs

We need to find the graph with vertex at \((5, 3)\) and opening upwards. Looking at the given graphs, the first graph (top one) has its vertex around \((5, 3)\) (checking the grid: x=5, y=3) and opens upwards, while the second graph's vertex seems to be at a different x - coordinate (not 5).

Answer:

The graph (the top one among the given options) with vertex at \((5, 3)\) and opening upwards. (If we consider the first option as the one with vertex at (5,3), then the answer is the first option's graph.)