Answer:
Explanation:
The vertex form of a quadratic function is given by the equation f(x) = a(x - h)² + k, where (h, k) is the coordinates of the vertex of the parabola, and a is a constant coefficient.
To find the values of a, h, and k for the given quadratic function f(x) = x² + 8x + 7, we can first use the formula for the x-coordinate of the vertex of a quadratic function, which is given by h = -b / 2a. In this case, the quadratic function is f(x) = x² + 8x + 7, so b = 8 and a = 1. Plugging these values into the formula, we get:
h = -8 / 2(1) = -4
Therefore, the value of h is -4.
Next, we can use the formula for the y-coordinate of the vertex of a quadratic function, which is given by k = f(h) = a(h - h)² + k. In this case, we know that the value of h is -4, and we can plug this value into the original quadratic function to get the value of k:
k = f(-4) = (-4)² + 8(-4) + 7 = 16 - 32 + 7 = -9
Therefore, the value of k is -9.
Finally, we can use the equation f(x) = a(x - h)² + k to find the value of a. Since we know that the coordinates of the vertex are (-4, -9) and the equation of the quadratic function is f(x) = x² + 8x + 7, we can plug these values into the equation to solve for a:
f(x) = a(x - h)² + k
x² + 8x + 7 = a(x - (-4))² + (-9)
We can then distribute the squared term and simplify the equation to get:
x² + 8x + 7 = a(x² - 8x + 16) + (-9)
x² + 8x + 7 = a(x²) - 8a(x) + 16a + (-9)
We can then collect like terms to get:
x² + 8x + 7 = a(x²) - 8ax + 16a - 9
Comparing the coefficients of the x², x, and constant terms on both sides of the equation, we can see that a = 1.
Therefore, the values of a, h, and k for the given quadratic function f(x) = x² + 8x + 7 are a = 1, h = -4, and k = -9.