How do you find the vertex of a quadratic function?
What are the steps to solving a quadratic function using factoring?
Please help!
http://www.uncwil.edu/courses/mat111hb/Pandr/quadratic/quadratic.html
Explain a process you would use to graph a quadratic equation. Include how to find the x-intercepts, the vertex, and the axis of symmetry.
To find the vertex of a quadratic function, you can follow these steps:
1. Start with a quadratic function in the form of f(x) = ax^2 + bx + c.
2. Identify the coefficients of the quadratic function: a, b, and c.
3. Use the formula for the x-coordinate of the vertex, given by: x = -b / (2a).
4. Substitute the value of x into the quadratic function to find the corresponding y-coordinate of the vertex.
Here's an example to illustrate the steps:
Given the quadratic function f(x) = 2x^2 + 4x + 1, we can find the vertex:
Step 1: Identify the coefficients of the quadratic function:
a = 2
b = 4
c = 1
Step 2: Use the formula for the x-coordinate of the vertex:
x = -b / (2a)
= -4 / (2 * 2)
= -4 / 4
= -1
Step 3: Substitute the x-coordinate into the quadratic function to find the y-coordinate:
f(-1) = 2(-1)^2 + 4(-1) + 1
= 2(1) - 4 + 1
= 2 - 4 + 1
= -1
Therefore, the vertex of the quadratic function f(x) = 2x^2 + 4x + 1 is (-1, -1).
Now, let's move on to solving a quadratic function using factoring.
To solve a quadratic function using factoring, you can follow these steps:
1. Write the quadratic function in the form of f(x) = ax^2 + bx + c.
2. Factor the quadratic expression on the right side of the equation.
3. Set each factor equal to zero and solve for x.
4. The solutions you find are the x-intercepts or roots of the quadratic function.
Here's an example to illustrate the steps:
Given the quadratic function f(x) = x^2 - 3x - 10, we can solve it using factoring:
Step 1: Write the quadratic function:
f(x) = x^2 - 3x - 10
Step 2: Factor the quadratic expression:
(x - 5)(x + 2) = 0
Step 3: Set each factor equal to zero and solve for x:
x - 5 = 0 --> x = 5
x + 2 = 0 --> x = -2
Step 4: The solutions are x = 5 and x = -2, which are the x-intercepts or roots of the quadratic function.
Therefore, the solutions to the quadratic function f(x) = x^2 - 3x - 10 are x = 5 and x = -2.