Y= -(x-5)(x-1)
Finding AOS, Vertex, Y- Intecept, X Intercept, and two additional points
Do you know what the sketch looks like?
It has a maximum at the vertex (because of the negative outside of the brackets).
You might want to start with the intercepts.
Set x = 0 and solve for y (that will be the y-intercept).
Then for the x intercept set y equal to 0 and solve for x : )
To find the Axis of Symmetry (AOS), Vertex, Y-Intercept, X-Intercept, and two additional points of the quadratic equation y = -(x - 5)(x - 1), we can follow these steps:
Step 1: Determine the AOS:
The AOS is the vertical line that passes through the vertex of the parabola. For a quadratic equation in the form y = ax^2 + bx + c, the formula to find the AOS is given by x = -b/2a.
In this case, the quadratic equation is y = -(x - 5)(x - 1). The coefficient of x^2 is -1, and the coefficient of x is also -1. Therefore, the AOS can be calculated as follows:
x = -(-1) / (2 * -1)
x = 1/2
So the AOS is x = 1/2.
Step 2: Find the Vertex:
The vertex is the point on the parabola that lies on the AOS. To find the vertex, substitute the x-value of the AOS into the equation and calculate the corresponding y-value.
Using x = 1/2, substituting:
y = -((1/2) - 5)((1/2) - 1)
y = -(-9/2)(-1/2)
y = (-9/2)(-1/2)
y = 9/4
Therefore, the vertex is (1/2, 9/4).
Step 3: Calculate the Y-Intercept:
The Y-Intercept is the point where the parabola intersects the y-axis. To find this point, substitute x = 0 into the equation, and calculate the corresponding y-value.
Using x = 0, substituting:
y = -(0 - 5)(0 - 1)
y = -(-5)(-1)
y = 5
Therefore, the Y-Intercept is (0, 5).
Step 4: Determine the X-Intercepts:
The X-Intercepts, also known as the roots or zeros, are the points where the parabola intersects the x-axis. To find these points, set y = 0 and solve for x by factoring or using the quadratic formula.
Setting y = 0 and factoring:
0 = -(x - 5)(x - 1)
Setting each factor equal to zero:
x - 5 = 0 --> x = 5
x - 1 = 0 --> x = 1
Therefore, the X-Intercepts are (5, 0) and (1, 0).
Step 5: Find two additional points:
To find two additional points, substitute different x-values into the equation and calculate the corresponding y-values.
For example, substituting x = 2:
y = -(2 - 5)(2 - 1)
y = -(3)(1)
y = -3
So one additional point is (2, -3).
Substituting x = 3:
y = -(3 - 5)(3 - 1)
y = -(2)(-2)
y = 4
So another additional point is (3, 4).
To summarize:
- Axis of Symmetry (AOS): x = 1/2
- Vertex: (1/2, 9/4)
- Y-Intercept: (0, 5)
- X-Intercepts: (5, 0) and (1, 0)
- Additional points: (2, -3) and (3, 4)