The y-intercept of a function is the point where the graph crosses the ![\mathbf{y = x^3 + 2x^2 - 193x - 270 }](https://tex.z-dn.net/?f=%5Cmathbf%7By%20%3D%20x%5E3%20%2B%202x%5E2%20-%20193x%20-%20270%20%7D)
- 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 }](https://tex.z-dn.net/?f=%5Cmathbf%7By%20%3D%20x%5E3%20%2B%202x%5E2%20-%20193x%20-%20270%20%7D)
<u>(a) The factors</u>
First, we write out the points where the function cross the x-axis.
The points are:
![\mathbf{x = 10}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%3D%2010%7D)
![\mathbf{x = -3}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%3D%20-3%7D)
![\mathbf{x = -9}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%3D%20-9%7D)
Equate the above points to 0
![\mathbf{x -10 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20-10%20%3D%200%7D)
![\mathbf{x +3 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%2B3%20%3D%200%7D)
![\mathbf{x +9 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%2B9%20%3D%200%7D)
Hence, the factors are: (x - 10), (x + 3) and (x + 9)
<u>(b) The y-intercept</u>
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
<u>(c) The standard form</u>
In (a), we have:
![\mathbf{x -10 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20-10%20%3D%200%7D)
![\mathbf{x +3 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%2B3%20%3D%200%7D)
![\mathbf{x +9 = 0}](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%20%2B9%20%3D%200%7D)
Multiply the above equations
![\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 \times 0 \times 0}](https://tex.z-dn.net/?f=%5Cmathbf%7B%28x%20-%2010%29%20%5Ctimes%20%28x%20%2B%203%29%20%5Ctimes%20%28x%20%2B%209%29%20%3D%200%20%5Ctimes%200%20%5Ctimes%200%7D)
![\mathbf{(x - 10) \times (x + 3) \times (x + 9) = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7B%28x%20-%2010%29%20%5Ctimes%20%28x%20%2B%203%29%20%5Ctimes%20%28x%20%2B%209%29%20%3D%200%20%7D)
Expand
![\mathbf{(x - 10) \times (x^2 + 3x + 9x + 27) = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7B%28x%20-%2010%29%20%5Ctimes%20%28x%5E2%20%2B%203x%20%2B%209x%20%2B%2027%29%20%3D%200%20%7D)
![\mathbf{(x - 10) \times (x^2 + 12x + 27) = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7B%28x%20-%2010%29%20%5Ctimes%20%28x%5E2%20%2B%2012x%20%2B%2027%29%20%3D%200%20%7D)
Expand
![\mathbf{x^3 + 12x^2 + 27x - 10x^2 - 220x - 270 = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%5E3%20%2B%2012x%5E2%20%2B%2027x%20-%2010x%5E2%20-%20220x%20-%20270%20%3D%200%20%7D)
Collect like terms
![\mathbf{x^3 + 12x^2 - 10x^2+ 27x - 220x - 270 = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%5E3%20%2B%2012x%5E2%20-%2010x%5E2%2B%2027x%20%20-%20220x%20-%20270%20%3D%200%20%7D)
![\mathbf{x^3 + 2x^2 - 193x - 270 = 0 }](https://tex.z-dn.net/?f=%5Cmathbf%7Bx%5E3%20%2B%202x%5E2%20-%20193x%20-%20270%20%3D%200%20%7D)
Replace 0 with y
![\mathbf{y = x^3 + 2x^2 - 193x - 270 }](https://tex.z-dn.net/?f=%5Cmathbf%7By%20%3D%20x%5E3%20%2B%202x%5E2%20-%20193x%20-%20270%20%7D)
Hence, the standard form is: ![\mathbf{y = x^3 + 2x^2 - 193x - 270 }](https://tex.z-dn.net/?f=%5Cmathbf%7By%20%3D%20x%5E3%20%2B%202x%5E2%20-%20193x%20-%20270%20%7D)
Read more about graphs and functions at:
brainly.com/question/18806107