Question - Quadratic Function Equation Calculation from Graph
Solution:
To find the equation of the quadratic function g whose graph is shown, we will use the vertex form of a quadratic function, which is:\[ g(x) = a(x - h)^2 + k \]where (h, k) is the vertex of the parabola, and 'a' is a coefficient that determines the width and direction (upward or downward) of the parabola. From the graph, we can see that the vertex of the parabola is at (-4, 2). Therefore, h = -4 and k = 2.To find 'a', we need another point on the graph. We can use the point (-5, 0), which is one of the x-intercepts. Now, substitute h, k, and the coordinates of the point (-5, 0) into the vertex form and solve for a:\[ 0 = a(-5 - (-4))^2 + 2 \]\[ 0 = a(-5 + 4)^2 + 2 \]\[ 0 = a(1)^2 + 2 \]\[ 0 = a + 2 \]\[ a = -2 \]So the coefficient 'a' is -2. Now we can write the complete equation of the function:\[ g(x) = -2(x - (-4))^2 + 2 \]\[ g(x) = -2(x + 4)^2 + 2 \]That is the equation of the quadratic function g whose graph is shown in the image.
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