Question

which of the following is the equation of the function shown below? graph with axes 0 - 10, parabola opening up options: a none of these; b ( y=(x + 2)^2 + 4 ); c ( y=(x - 4)^2 + 2 ); d ( y=(x - 2)^2 + 8 ); e ( y=(x + 4)^2 + 2 ); f ( y=(x + 2)^2 + 8 ); g ( y=(x - 2)^2 + 4 )

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.

Step2: Identify the vertex from the graph

From the graph, the vertex (the minimum point) seems to be at \((2, 4)\) (since it's around x=2 and y=4).

Step3: Substitute h and k into vertex form

Using \( h = 2 \) and \( k = 4 \), the equation should be \( y=(x - 2)^2+4 \), which is option G.

Step4: Verify with y-intercept (optional)

Check when \( x = 0 \), \( y=(0 - 2)^2+4=4 + 4 = 8 \)? Wait, no, the graph at x=0 has y around 8? Wait, maybe my initial vertex estimate was off. Wait, let's recheck. Wait, the options: let's check each option's vertex.

Option G: \( y=(x - 2)^2+4 \), vertex (2,4). Let's see the graph: the parabola is opening upwards, vertex around (2,4)? Wait, the y-intercept: when x=0, y=( - 2)^2 +4=8. The graph at x=0 has y around 8? Wait, the graph shows at x=0, y is between 5 and 10, maybe 8. Then let's check other options.

Option B: \( y=(x + 2)^2+4 \), vertex (-2,4). But the graph's vertex is at positive x, so B is out.

Option C: \( y=(x - 4)^2+2 \), vertex (4,2). The graph's vertex is at x=2, so C is out.

Option D: \( y=(x - 2)^2+8 \), vertex (2,8). The graph's y at vertex is lower, so D is out.

Option E: \( y=(x + 4)^2+2 \), vertex (-4,2). Out.

Option F: \( y=(x + 2)^2+8 \), vertex (-2,8). Out.

Option G: \( y=(x - 2)^2+4 \), vertex (2,4). Wait, but when x=0, y=8. Wait, maybe the graph's y-intercept is 8? Let's see the graph: the y-axis has 0,5,10. The parabola crosses y-axis at around 8? Then x=0, y=8: for G, x=0, y=4 + 4=8. Yes! So that matches. So the vertex is (2,4), so the equation is \( y=(x - 2)^2+4 \), which is option G. Wait, but in the options, G is \( y=(x - 2)^2+4 \)? Wait the last option is G: \( y=(x - 2)^2+4 \). Yes. So that's the correct one.

Answer:

G. \( y=(x - 2)^2+4 \)