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: Identify the vertex form

The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the parent function \( f(x) = x^2 \), the vertex is \((0, 0)\). We need to find the vertex of \( g(x) \) from the graph.

Looking at the graph, the vertex (the minimum point) of \( g(x) \) is at \((5, -5)\)? Wait, no, let's check the grid again. Wait, the parabola crosses the x - axis at \( x = 2 \) and \( x = 6 \). The x - coordinate of the vertex (h) is the midpoint of the roots. The midpoint of 2 and 6 is \( \frac{2 + 6}{2}=\frac{8}{2}=4 \). Now, let's find the y - coordinate (k). Let's take a point on the parabola. When \( x = 4 \), looking at the graph, the y - value is - 5? Wait, no, let's check the graph again. Wait, the parabola has its vertex at (4, - 5)? Wait, no, maybe I made a mistake. Wait, the parent function is \( f(x)=x^{2}\), which has vertex (0,0). The transformed function \( g(x) \) is a translation, so \( a = 1 \) (since it's a translation, the vertical stretch/compression factor \( a \) is 1). Now, let's find the vertex. The roots are at \( x = 2 \) and \( x = 6 \), so the axis of symmetry is \( x=\frac{2 + 6}{2}=4 \). So the x - coordinate of the vertex \( h = 4 \). Now, let's find the y - coordinate. Let's plug \( x = 4 \) into the equation? Wait, no, let's use the fact that for \( f(x)=x^{2}\), when we translate it \( h \) units horizontally and \( k \) units vertically, the function is \( g(x)=(x - h)^{2}+k \). The vertex is (h,k). From the graph, the vertex is at (4, - 5)? Wait, no, let's check the graph again. Wait, the parabola is opening upwards, same as \( f(x)=x^{2}\), so \( a = 1 \). The vertex is at (4, - 5)? Wait, maybe I miscalculated. Wait, when \( x = 4 \), the y - value is - 5? Let's check the grid. The vertical axis: each grid line is 1 unit. The vertex is at (4, - 5)? Wait, no, let's take another approach. Let's use the vertex form. Since the parabola is a translation of \( y=x^{2}\), \( a = 1 \). The vertex (h,k) can be found from the graph. Looking at the graph, the lowest point (vertex) is at (4, - 5)? Wait, no, maybe the vertex is at (4, - 5)? Wait, let's check the roots. The parabola intersects the x - axis at \( x = 2 \) and \( x = 6 \). So the equation of the parabola can be written as \( g(x)=(x - 2)(x - 6) \). Let's expand this: \( g(x)=x^{2}-8x + 12 \). Now, let's convert this to vertex form. Completing the square: \( x^{2}-8x+12=(x^{2}-8x + 16)-16 + 12=(x - 4)^{2}-4 \). Wait, that's different. Wait, maybe my initial root finding was wrong. Wait, looking at the graph, when \( x = 0 \), \( y = 12 \)? Wait, no, the graph crosses the y - axis? Wait, no, the graph is a parabola opening upwards, with roots at \( x = 2 \) and \( x = 6 \), and vertex at (4, - 4)? Wait, let's recalculate. \( (x - 2)(x - 6)=x^{2}-8x + 12 \). Completing the square: \( x^{2}-8x= (x - 4)^{2}-16 \), so \( x^{2}-8x + 12=(x - 4)^{2}-16 + 12=(x - 4)^{2}-4 \). Ah, so the vertex is (4, - 4). So \( h = 4 \), \( k=-4 \), and \( a = 1 \). So the function is \( g(x)=(x - 4)^{2}-4 \).

Step2: Confirm the vertex form

Since the function is a translation of \( f(x)=x^{2}\), the value of \( a \) is 1 (no vertical stretch or compression, just a horizontal and vertical translation). The vertex of \( g(x) \) is at (4, - 4), so \( h = 4 \) and \( k=-4 \). Plugging these into the vertex form \( g(x)=a(x - h)^{2}+k \), with \( a = 1 \), we get \( g(x)=(x - 4)^{2}-4 \).

Answer:

\( g(x)=(x - 4)^{2}-4 \)