Question

Write the polynomial function of the graph below. Leave your function in factored form. (-3,0)(2.25,68)(0,0)(1,0.75)(2,0)(3,40)

Answer 1

To write the polynomial function based on the given points, we will use the x-intercepts and analyze the additional points provided.

1. **Identify the x-intercepts**: From the points given, the x-intercepts are at:
- \( x = -3 \)
- \( x = 0 \)
- \( x = 1 \)
- \( x = 2 \)

2. **Create factors for each x-intercept**:
- For \( x = -3 \), the factor is \( (x + 3) \)
- For \( x = 0 \), the factor is \( x \)
- For \( x = 1 \), the factor is \( (x - 1) \)
- For \( x = 2 \), the factor is \( (x - 2) \)

3. **Multiply the factors together** to form the polynomial:
\[
f(x) = k(x + 3)(x)(x - 1)(x - 2)
\]
where \( k \) is a constant multiplier that we can find using another point from the graph.

4. **Use another point to find \( k \)**: We can use the point \( (2.25, 68) \) to determine \( k \).

Substitute \( x = 2.25 \) into the polynomial:
\[
f(2.25) = k(2.25 + 3)(2.25)(2.25 - 1)(2.25 - 2)
\]
Calculate the components:
- \( 2.25 + 3 = 5.25 \)
- \( 2.25 - 1 = 1.25 \)
- \( 2.25 - 2 = 0.25 \)

Now substituting:
\[
f(2.25) = k (5.25)(2.25)(1.25)(0.25) = 68
\]

Calculate \( (5.25)(2.25)(1.25)(0.25) \):
- Calculate step by step:
- \( 5.25 \times 2.25 = 11.8125 \)
- \( 11.8125 \times 1.25 = 14.765625 \)
- \( 14.765625 \times 0.25 = 3.69140625 \)

Now we have:
\[
k \cdot 3.69140625 = 68
\]
Solving for \( k \):
\[
k = \frac{68}{3.69140625} \approx 18.41
\]

5. **Final Polynomial in Factored Form**:
The polynomial function in factored form is:
\[
f(x) \approx 18.41 (x + 3)(x)(x - 1)(x - 2)
\]

Thus, the final answer for the polynomial function in factored form is approximately
\[
f(x) = 18.41 (x + 3)(x)(x - 1)(x - 2).
\]

(Note: If you need \( k \) in its exact form, it can be computed exactly based on the earlier calculations for precise numerical results).