Question

the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x).
write your answer in the form a(x - h)² + k, where a, h, and k are integers or simplified fractions.
g(x) =

Explanation:

Step1: Find the vertex of \( g(x) \)

The vertex of a parabola in the form \( a(x - h)^2 + k \) is \( (h, k) \). From the graph, the vertex of \( g(x) \) is at \( (-3, -1) \)? Wait, no, looking at the graph, the minimum point (vertex) is between \( x = -4 \) and \( x = -2 \), actually, the vertex is at \( (-3, -1) \)? Wait, no, let's check the grid. The parabola \( f(x)=x^2 \) has vertex at \( (0,0) \). The graph of \( g(x) \) has its vertex at \( (-3, -1) \)? Wait, no, looking at the graph, the vertex is at \( (-3, -1) \)? Wait, no, let's see the x-intercepts? Wait, the graph touches the x-axis at \( x = -4 \) and \( x = -2 \)? Wait, no, the parabola crosses the x-axis? No, it's a minimum point. Wait, the vertex form of a parabola is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the graph, the vertex is at \( (-3, -1) \)? Wait, no, let's check the coordinates. Wait, the original function is \( f(x)=x^2 \), which has vertex \( (0,0) \). The graph of \( g(x) \) is a translation. Let's find the vertex: looking at the graph, the lowest point (vertex) is at \( x = -3 \), \( y = -1 \)? Wait, no, maybe I made a mistake. Wait, the graph of \( g(x) \) has its vertex at \( (-3, -1) \)? Wait, no, let's count the grid. The vertex is at \( (-3, -1) \)? Wait, no, let's see the y-intercept. Wait, when \( x = 0 \), \( g(0) \) is 8? Wait, no, the graph passes through \( (0, 8) \)? Wait, maybe the vertex is at \( (-3, -1) \)? Wait, no, let's use the vertex form. The general form of a translated parabola from \( f(x)=x^2 \) is \( g(x)=(x - h)^2 + k \), since there's no vertical stretch (a=1, because the shape is same as \( x^2 \)). Wait, the vertex of \( g(x) \): let's find the midpoint of the two x-intercepts (if it's a parabola opening upwards). Wait, the graph intersects the x-axis at \( x = -4 \) and \( x = -2 \)? No, it's a minimum point. Wait, the vertex is at \( (-3, -1) \)? Wait, no, let's calculate. The vertex is at \( h = \frac{-4 + (-2)}{2} = -3 \), and \( k \) is the y-coordinate of the vertex. From the graph, when \( x = -3 \), \( y = -1 \)? Wait, no, maybe the vertex is at \( (-3, -1) \). Wait, but let's check with the point \( (0, 8) \). Wait, if \( g(x)=(x + 3)^2 - 1 \), then \( g(0)=(3)^2 - 1 = 9 - 1 = 8 \), which matches the y-intercept (since when \( x = 0 \), \( y = 8 \)). So the vertex is at \( (-3, -1) \), so \( h = -3 \), \( k = -1 \), and \( a = 1 \) (since the parabola is not stretched or compressed, same as \( x^2 \)). Wait, let's verify. So \( g(x)=(x - (-3))^2 + (-1) = (x + 3)^2 - 1 \). Let's expand it: \( (x + 3)^2 - 1 = x^2 + 6x + 9 - 1 = x^2 + 6x + 8 \). Let's check \( x = -4 \): \( (-4)^2 + 6(-4) + 8 = 16 - 24 + 8 = 0 \), which matches the x-intercept. \( x = -2 \): \( (-2)^2 + 6(-2) + 8 = 4 - 12 + 8 = 0 \), which also matches. And \( x = 0 \): \( 0 + 0 + 8 = 8 \), which matches the y-intercept. So the vertex is at \( (-3, -1) \), so \( g(x)=(x + 3)^2 - 1 \).

Step2: Write the function rule

Since the vertex is \( (-3, -1) \) and \( a = 1 \) (no vertical stretch), the function rule is \( g(x)=(x + 3)^2 - 1 \).

Answer:

\( g(x)=(x + 3)^2 - 1 \)