the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x). \nwrite your…

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 )