Question

Use the factors of the function and the y-intercept to find the standard form of the equation representing Jared’s function. Type the correct answer in the box.

Answer 1

The y-intercept of a function is the point where the graph crosses the $\mathbf{y = x^3 + 2x^2 - 193x - 270 }$

• The factors of the Jared's graph are: (x - 10), (x + 3) and (x + 9)
• The y-intercept is -13.5
• The standard equation of the function is: $\mathbf{y = x^3 + 2x^2 - 193x - 270 }$

(a) The factors

First, we write out the points where the function cross the x-axis.

The points are:

$\mathbf{x = 10}$

$\mathbf{x = -3}$

$\mathbf{x = -9}$

Equate the above points to 0

$\mathbf{x -10 = 0}$

$\mathbf{x +3 = 0}$

$\mathbf{x +9 = 0}$

Hence, the factors are: (x - 10), (x + 3) and (x + 9)

(b) The y-intercept

This is the point where the graph crosses the y-axis.

From the attached graph, the graph crosses the y-axis at -13.5.

Hence, the y-intercept is -13.5

(c) The standard form

In (a), we have:

$\mathbf{x -10 = 0}$

$\mathbf{x +3 = 0}$

$\mathbf{x +9 = 0}$

Multiply the above equations

$\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 \times 0 \times 0}$

$\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 }$

Expand

$\mathbf{(x - 10) \times (x^2 + 3x + 9x + 27) = 0 }$

$\mathbf{(x - 10) \times (x^2 + 12x + 27) = 0 }$

Expand

$\mathbf{x^3 + 12x^2 + 27x - 10x^2 - 220x - 270 = 0 }$

Collect like terms

$\mathbf{x^3 + 12x^2 - 10x^2+ 27x - 220x - 270 = 0 }$

$\mathbf{x^3 + 2x^2 - 193x - 270 = 0 }$

Replace 0 with y

$\mathbf{y = x^3 + 2x^2 - 193x - 270 }$

Hence, the standard form is: $\mathbf{y = x^3 + 2x^2 - 193x - 270 }$

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