To find the vertex form of the function, we complete the square.
s(x) = x^2 - 8x + 7
First, we need to factor out the coefficient of the x^2 term, which is 1.
s(x) = (x^2 - 8x) + 7
Next, take half of the coefficient of the x term, which is -8 (-8/2 = -4), and square it (16).
s(x) = (x^2 - 8x + 16) + 7 - 16
Now, rewrite the expression inside parentheses as a binomial squared.
s(x) = (x - 4)^2 + 7 - 16
Simplify the expression within the parentheses.
s(x) = (x - 4)^2 - 9
Therefore, the vertex form of the function is s(x) = (x - 4)^2 - 9.
(A) To find the x-intercepts, we set s(x) = 0 and solve for x.
0 = (x - 4)^2 - 9
(x - 4)^2 = 9
Take the square root of both sides.
x - 4 = ±3
Solve for x.
x = 4 + 3 or x = 4 - 3
x = 7 or x = 1
The x-intercepts are (7, 0) and (1, 0).
(B) The vertex of the function is found by taking the opposite of the x-coordinate of the binomial squared and the y-coordinate is the constant term outside the parentheses.
Vertex = (4, -9)
(C) Since the coefficient of the x^2 term is positive (positive 1 in this case), the parabola opens upwards, which means the vertex is a minimum point. The minimum value of the function is the y-coordinate of the vertex.
Minimum value = -9
(D) The range of the function is all real numbers greater than or equal to the minimum value.
Range = y ≥ -9
Find the vertex form of the function. Then find each of the following.
(A) Intercepts
(B) Vertex
(C) Maximum or minimum and what are the values
(D) Range
s(x)= x^2-8x+7
1 answer