Basic High School Math

Question

Sketch a function with the following characteristics (note: there are many possible correct functions). For full credit, each piece below must be represented and the function’s end behavior must be correct (1 point for each correct item, including correct overall shape).

1. y-intercept: -3
2. Roots: -4, 1, 8
3. Relative minimum at (-2, -4)
4. Relative maximum at (6, 5)
5. As𝑥→−∞,𝑦→∞
6. As𝑥→∞,𝑦→−∞

Answer 1

a(x+4)(x-1)(x-8) = a(x^2 +3x -4)(x-8) = a(x^3 -5x^2 -28x +32)

to get a y intercept of -3 set a= -3/32

f(x) = -3x^3/32 + 15x^2/32 + 140x/32 - 330

that gets you 5 out of 7 points, satisfying the required roots, y intercept and end behavior

but no points on the local extrema

f'(x) = -9x^2/32 +30x/32 + 140/32 = 0

-9x^2 + 30x +140 = 0

9x^2 -30x - 140 = 0

which doesn't have the necessary x values for the given local max and min

(x-6)(x+2) = 0

x = -2 or 6

just an offhand guess, but you may need a higher degree function to get all 7 points. maybe add in some more factors, but otherwise keep the rest relatively the same.