Use the following steps to create a graph of 4x3−8x2−9x+18 . Factor the polynomial by grouping. One of the factors is quadratic. Factor it into two linear factors. Does it match a factoring pattern you’ve seen before using a polynomial identity? The polynomial can be factored into three linear factors. Use those factors to find the zeros of the function. Number the x-axis from −2 to 4, with intervals of 0.5. Number the y-axis from −20 to 40, with intervals of 5. Place three dots on the x -axis corresponding to the zeros you have found. You will be drawing a curve that passes through those three dots. Will the curve fall on the left and rise on the right, or rise on the left and fall on the right? How can you tell? The curve will have a relative maximum at approximately (−0.5,20) . The curve will have a relative minimum at approximately (1.75,−1) . Place dots at those locations. Now draw a smooth curve to represent the polynomial. The curve should have the proper end behavior on the left and right and pass through the three dots on the x-axis, with a relative maximum and minimum at the points you have located. Compare your results with the curve produced by a graphing calculator. Fill in the following with the corresponding x-values. The domain of the function goes from to . Fill in the following with the corresponding y-values. The range of the function goes from to . Fill in the following intervals with the corresponding x -values: The curve is negative from to and from to . The curve is positive from to and from to . Fill in the following intervals with the corresponding x -values: The curve is rising from to and from to . The curve is falling from to . Fill in the following with the average rates of change. The average rate of change over the interval [−2,0] is . The average rate of change over the interval [−3,4] is .

Question

GPT-4o mini · Answer

To factor the polynomial $ f(x) = 4x^3 - 8x^2 - 9x + 18 $ by grouping, we proceed as follows: 1. **Factor by Grouping:** - Group the terms: $ (4x^3 - 8x^2) + (-9x + 18) $ - Factor out the common factors from each group: - From $ 4x^3 - 8x^2 $, we can factor out $ 4x^2 $: $$ 4x^2(x - 2) $$ - From $ -9x + 18 $, we can factor out $ -9 $: $$ -9(x - 2) $$ - We can now factor out the common binomial $ x - 2 $: $$ f(x) = (x - 2)(4x^2 - 9) $$ 2. **Factor the Quadratic:** - The quadratic $ 4x^2 - 9 $ is a difference of squares: $$ 4x^2 - 9 = (2x - 3)(2x + 3) $$ - Therefore, the complete factorization of the polynomial is: $$ f(x) = (x - 2)(2x - 3)(2x + 3) $$ 3. **Find the Zeros:** - Set $ f(x) = 0 $: $$ (x - 2)(2x - 3)(2x + 3) = 0 $$ - The zeros are: 1. $ x - 2 = 0 $ ⇒ $ x = 2 $ 2. $ 2x - 3 = 0 $ ⇒ $ x = \frac{3}{2} = 1.5 $ 3. $ 2x + 3 = 0 $ ⇒ $ x = -\frac{3}{2} = -1.5 $ - Thus, the zeros are $ x = 2, 1.5, -1.5 $. 4. **Graphing Information:** - The graph has zeros at $ x = 2, 1.5, -1.5 $, represented as dots on the x-axis at these points. - For end behavior: - The leading coefficient (4) is positive, and the degree of the polynomial is odd (3), so the curve falls on the left and rises on the right. - The relative maximum is at approximately $ (-0.5, 20) $, and the relative minimum is at approximately $ (1.75, -1) $. 5. **Domain and Range:** - The domain of the function is: $$ (-\infty, \infty) $$ - The range is also: $$ (-\infty, \infty) $$ (The polynomial can take all y values since it is cubic and unbounded). 6. **Intervals for Positivity and Negativity:** - The curve is negative between the zeros, specifically: - From $ -1.5 $ to $ 1.5 $ (negative between the first zero -1.5 and second zero 1.5) and also from $ -\infty $ to $ -1.5 $. - Positive intervals are: - From $ (-\infty, -1.5) $ (down to -1.5) and from $ (1.5, 2) $ (coming back up to 2). 7. **Intervals for Increasing and Decreasing:** - The curve is rising: - From $ (-\infty, -1.5) $ and from $ (1.5, 2) $. - The curve is falling: - From $ (-1.5, 1.5) $. 8. **Average Rates of Change:** - To find the average rate of change over the intervals: $-2$ to $0$: $$ \text{Average Rate of Change} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{18 - (f(-2))}{2} $$ where $ f(-2) = 4(-2)^3 - 8(-2)^2 - 9(-2) + 18 = -32 - 32 + 18 + 18 = -28 $. Thus: $$ \text{Average Rate of Change} = \frac{18 - (-28)}{2} = \frac{46}{2} = 23 $$ - Over the interval $[-3, 4]$: $$ \text{Average Rate of Change} = \frac{f(4) - f(-3)}{4 - (-3)} = \frac{f(4) - f(-3)}{7} $$ Calculate $ f(4) = 4(4^3) - 8(4^2) - 9(4) + 18 = 64 - 128 - 36 + 18 = -82 $ and $ f(-3) = -28 $. Thus: $$ \text{Average Rate of Change} = \frac{-82 - (-28)}{7} = \frac{-82 + 28}{7} = \frac{-54}{7} \approx -7.71 $$ 9. **Summary:** - **Zeros:** $ x = 2, 1.5, -1.5 $ - **Domain:** $ (-\infty, \infty) $ - **Range:** $ (-\infty, \infty) $ - **Negative Intervals:** From $ -1.5 $ to $ 1.5 $ - **Positive Intervals:** From $ (-\infty, -1.5) $ and from $ (1.5, 2) $ - **Rising Intervals:** From $ (-\infty, -1.5) $ and $ (1.5, 2) $ - **Falling Intervals:** From $ (-1.5, 1.5) $ - **Average Rate of Change:** - Over $ [-2, 0] = 23 $ - Over $ [-3, 4] \approx -7.71 $ You can verify the curve by graphing it using a graphing calculator and comparing it with your drawn graph.