Chapter Closure
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You will need to substitute 0 and solve for a variable twice before graphing.
y-intercept: (0,- 15)
x-intercepts: (- 3,0) and (5,0)
To determine the x- and y-intercepts of a line, we need to substitute 0 for one variable, solve, then repeat for the other variable.
To find where a function intercepts the x-axis, the function can be set equal to zero. Then, the x-values that satisfy the equation are the zeros of the function, also called the roots. Thus, to find the x-intercepts of the given function we can set it equal to zero and solve for x. x^2-2x-15=0 To solve the above equation for x, we can start by factoring. Then we will use the Zero Product Property.
x^2-2x- 15=0 In this case we have - 15. This is a negative number, so for the product of the constant terms in the factors to be negative, these constants must have the opposite sign — one positive and one negative.
| Factor Constants | Product of Constants |
|---|---|
| 1 and - 15 | - 15 |
| - 1 and 15 | - 15 |
| -3 and 5 | - 15 |
| 3 and - 5 | - 15 |
Next, let's consider the coefficient of the linear term. x^2-2x-15=0 For this term, we need the sum of the factors that produced the constant term to equal the coefficient of the linear term, - 2.
| Factors | Sum of Factors |
|---|---|
| 1 and - 15 | - 14 |
| -1 and 15 | 14 |
| 3 and - 5 | - 2 |
| - 3 and 5 | 2 |
We found the factors whose product is - 15 and whose sum is - 2. x^2- 2x-15=0 ⇕ (x+3)(x-5)=0 Now, the equation is written in a factored form.
Use the Zero Product Property
(I): LHS-3=RHS-3
(II): LHS+5=RHS+5
x= 0
Calculate power
Zero Property of Multiplication