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) There is no y-intercept. vertex: (4, -16)
C. x-intercept: (0, 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) y-intercept: (0, 0) vertex: (4, -16)
To find the key points on the graph of the equation y = x^2 - 8x, we can use algebraic methods to solve for the x-intercepts and y-intercept, as well as find the coordinates of the vertex.
1. X-intercepts: To find the x-intercepts, we set y = 0 and solve for x.
Set x^2 - 8x = 0, and factor out x to get x(x - 8) = 0.
Set each factor equal to zero: x = 0 and x - 8 = 0.
The x-intercepts are at (0, 0) and (8, 0).
2. Y-intercept: To find the y-intercept, we set x = 0 and solve for y.
Substitute x = 0 into the equation y = x^2 - 8x: y = 0^2 - 8(0) = 0.
The y-intercept is at (0, 0).
3. Vertex: The x-coordinate of the vertex can be found using the formula x = -b/(2a).
In this case, a = 1 and b = -8. Substituting those values, we get x = -(-8)/(2*1) = 8/2 = 4.
To find the y-coordinate, substitute the x-coordinate into the original equation:
y = (4)^2 - 8(4) = 16 - 32 = -16.
The vertex is at (4, -16).
Comparing the answers given, we can see that the correct option is:
C. x-intercept: (0, 0) y-intercept: (0, 0) vertex: (4, -16)
To determine the key points on the graph of y = x^2 - 8x, we can start by finding the x-intercepts. The x-intercepts occur when y = 0, so we need to solve the equation x^2 - 8x = 0.
Factoring out an x, we have: x(x - 8) = 0
This equation is satisfied when x = 0 or x - 8 = 0.
Therefore, the x-intercepts are (0, 0) and (8, 0).
Next, let's find the y-intercept. The y-intercept occurs when x = 0, so we substitute x = 0 into the original equation:
y = (0)^2 - 8(0) = 0 - 0 = 0
Therefore, the y-intercept is (0, 0).
Finally, to find the vertex, we can use the formula x = -b / (2a), where the equation is in the form y = ax^2 + bx + c.
For the given equation y = x^2 - 8x, a = 1, b = -8.
Plugging these values into the formula, we have:
x = -(-8) / (2 * 1) = 8 / 2 = 4
To find the y-coordinate of the vertex, substitute x = 4 into the original equation:
y = (4)^2 - 8(4) = 16 - 32 = -16
Therefore, the vertex is (4, -16).
Based on this analysis, the correct answer is option B: x-intercept: (0, 0) There is no y-intercept. vertex: (4, -16).