Create a graph; then answer the following question.
What are the key points on the graph of y = x^2 - 8x? Name the vertex, x-intercept(s), and y-intercept.
A. x-intercept: (4, -16) y-intercept: (0, 0) vertex: (8, 0)
B. x-intercept: (0, 0)(0, 8) There is no y-intercept. vertex: (4, -16)
C. x-intercept: (0, 0)(8, 0) y-intercept: (0, 0) vertex: (4, -16)
D. x-intercept: (8, 0) y-intercept: (0, 0) vertex: (4, -16)
C. x-intercept: (0, 0)(8, 0) y-intercept: (0, 0) vertex: (4, -16)
To create the graph of y = x^2 - 8x, we can start by finding the x-intercepts, y-intercept, and vertex of the equation.
1. X-intercepts: To find the x-intercepts, we set y = 0 and solve for x. So we have:
0 = x^2 - 8x
Using factoring or the quadratic formula, we find that the x-intercepts are (0, 0) and (8, 0).
2. Y-intercept: To find the y-intercept, we set x = 0 and solve for y. So we have:
y = (0)^2 - 8(0)
y = 0
Thus, the y-intercept is (0, 0).
3. Vertex: The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the formula:
Vertex x-coordinate, h = -b / (2a)
Vertex y-coordinate, k = f(h), where f is the function.
In our equation, y = x^2 - 8x, the coefficient of x^2 is 1 (a = 1) and the coefficient of x is -8 (b = -8), so we can find the vertex as follows:
h = -(-8) / (2 * 1) = 8 / 2 = 4
k = (4)^2 - 8(4) = 16 - 32 = -16
Thus, the vertex is (4, -16).
Now, let's compare the key points on the graph provided in the answer choices:
A. x-intercept: (4, -16) y-intercept: (0, 0) vertex: (8, 0)
B. x-intercept: (0, 0)(0, 8) There is no y-intercept. vertex: (4, -16)
C. x-intercept: (0, 0)(8, 0) y-intercept: (0, 0) vertex: (4, -16)
D. x-intercept: (8, 0) y-intercept: (0, 0) vertex: (4, -16)
Among the given options, the correct answer is option C.
So, the key points on the graph of y = x^2 - 8x are:
x-intercepts: (0, 0) and (8, 0)
y-intercept: (0, 0)
vertex: (4, -16)
To find the key points on the graph of y = x^2 - 8x, we can start by determining the x-intercepts.
To find the x-intercepts, we set y = 0 in the equation and solve for x:
0 = x^2 - 8x
Factoring out x:
0 = x(x - 8)
Setting each factor equal to 0:
x = 0 or x - 8 = 0
x = 0 or x = 8
So, the x-intercepts are (0, 0) and (8, 0).
Next, we can find the y-intercept by setting x = 0 in the equation:
y = (0)^2 - 8(0)
y = 0
Therefore, the y-intercept is (0, 0).
To find the vertex, we can use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation:
Here, a = 1 and b = -8.
x = -(-8) / (2*1)
x = 8 / 2
x = 4
To find the y-coordinate of the vertex, we substitute x = 4 back into the equation:
y = (4)^2 - 8(4)
y = 16 - 32
y = -16
So, the vertex is (4, -16).
The correct answer that corresponds to these key points is option D:
x-intercept: (8, 0)
y-intercept: (0, 0)
vertex: (4, -16)