Question

1. convert each function into intercept form. (complete 2/3)

a. ( m(x) = -x^2 - 5x + 36 )
b. ( y = (x - 1)^2 - 4 )
c. ( f(x) = 5x^2 - 30x - 80 )

Explanation:

Part a: \( m(x) = -x^2 - 5x + 36 \)

The intercept form of a quadratic function is \( y = a(x - p)(x - q) \), where \( p \) and \( q \) are the x-intercepts. First, we factor the quadratic.

Step 1: Multiply by -1 to make the leading coefficient positive

\( m(x) = - (x^2 + 5x - 36) \)

Step 2: Factor the quadratic inside the parentheses

We need two numbers that multiply to -36 and add to 5. Those numbers are 9 and -4.
\( x^2 + 5x - 36 = (x + 9)(x - 4) \)

Step 3: Write in intercept form

\( m(x) = - (x + 9)(x - 4) \) or \( m(x) = (-1)(x + 9)(x - 4) \)

Part b: \( y = (x - 1)^2 - 4 \)

This is a difference of squares, since \( (x - 1)^2 - 4 = (x - 1)^2 - 2^2 \). The formula for difference of squares is \( a^2 - b^2 = (a - b)(a + b) \).

Step 1: Apply difference of squares

Let \( a = x - 1 \) and \( b = 2 \). Then:
\( y = (x - 1 - 2)(x - 1 + 2) \)

Step 2: Simplify the factors

\( y = (x - 3)(x + 1) \)

Part c: \( f(t) = 5t^2 - 30t - 80 \)

First, factor out the greatest common factor (GCF), then factor the remaining quadratic.

Step 1: Factor out the GCF (5)

\( f(t) = 5(t^2 - 6t - 16) \)

Step 2: Factor the quadratic inside the parentheses

We need two numbers that multiply to -16 and add to -6. Those numbers are -8 and 2.
\( t^2 - 6t - 16 = (t - 8)(t + 2) \)

Step 3: Write in intercept form

\( f(t) = 5(t - 8)(t + 2) \)

Final Answers:

a. \( \boldsymbol{m(x) = - (x + 9)(x - 4)} \)
b. \( \boldsymbol{y = (x - 3)(x + 1)} \)
c. \( \boldsymbol{f(t) = 5(t - 8)(t + 2)} \)

Answer:

First, factor out the greatest common factor (GCF), then factor the remaining quadratic.

Step 1: Factor out the GCF (5)

\( f(t) = 5(t^2 - 6t - 16) \)

Step 2: Factor the quadratic inside the parentheses

We need two numbers that multiply to -16 and add to -6. Those numbers are -8 and 2.
\( t^2 - 6t - 16 = (t - 8)(t + 2) \)

Step 3: Write in intercept form

\( f(t) = 5(t - 8)(t + 2) \)

Final Answers:

a. \( \boldsymbol{m(x) = - (x + 9)(x - 4)} \)
b. \( \boldsymbol{y = (x - 3)(x + 1)} \)
c. \( \boldsymbol{f(t) = 5(t - 8)(t + 2)} \)