Quadratic Function Equation Calculation from Graph
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.
