To find the x and y intercepts and the coordinates of the vertex for the quadratic function -x^2 - 5x, you can follow these steps:
1. X-Intercepts (Roots):
Set the function equal to zero and solve for x:
-x^2 - 5x = 0
Factor out a -x from the equation:
-x(x + 5) = 0
Set each factor equal to zero:
-x = 0 and x + 5 = 0
Solving for x in each equation, you get:
For -x = 0, x = 0
For x + 5 = 0, x = -5
So, the x-intercepts are x = 0 and x = -5.
2. Y-Intercept:
To find the y-intercept, plug in x = 0 into the function:
-x^2 - 5x = -(0^2) - 5(0) = 0
The y-intercept is y = 0.
3. Vertex:
To find the vertex, you can use the formula for the x-coordinate of the vertex in a quadratic function, which is -b/2a, where 'a' is the coefficient of the x^2 term and 'b' is the coefficient of the x term. In this case, a = -1 and b = -5.
x-coordinate of the vertex = -(-5) / (2 * -1) = 5 / 2 = 2.5
Now, plug this x-coordinate back into the original function to find the y-coordinate of the vertex:
-x^2 - 5x = -(2.5^2) - 5(2.5) = -6.25 - 12.5 = -18.75
The vertex has coordinates (2.5, -18.75).
So, to summarize:
- X-intercepts: x = 0 and x = -5
- Y-intercept: y = 0
- Vertex coordinates: (2.5, -18.75)