Question

For this problem, you will be working with the function h(x)=−13x2(5x+10)(x−3) .(3 points)Step 1: Find the zeros of the function h(x) . Make sure to show your work for finding each zero.Write each zero as an ordered pair (x,y) . Do not include spaces in your answers.First Zero: Second Zero: Third Zero: Question 2Step 2: To help us determine the shape of the graph, we can find additional points in between each zero to add to our graph. Find the values below to help determine the shape of the graph in between the zeros. Don't forget to show your work for finding each value.(2 points)Write each answer in its simplest form. Express non-integers as reduced fractions.h(−1) = h(1) = Question 3Part 3: You should now have three zeros as well as two points between those zeros for a total of 5 points for your graph. You are now going to use those 5 points to create a graph of your function. Use the link below to access a Desmos template where you will enter those 5 points into a table. Desmos will use this table to generate a function of best fit. The graph should go through all 5 of your points. Taadaa!In the second box, you will see a place to enter your name. Please enter your first and last name. In the top right-hand corner of your screen, you will see a "share graph" button. Click on this button and choose "export image." Save your image to upload with your handwritten work at the end of this portfolio.Desmos Link: 3.7 Graphs of Polynomials Portfolio Template Part 4: End BehaviorUse the graph Desmos graphed for you in part 3 and the original function, h(x)=−13x2(5x+10)(x−3) , to answer the following questions.(1 point)What is the degree of h(x) ? What is the leading coefficient of h(x) ? Question 4(3 points)As x approaches −∞ , h(x) approaches , and as x approaches +∞ , h(x) approaches .Analyzing the equation of the function, h(x) , helps me determine the end behavior because

Answer 1

Step 1 — zeros (showing work)
h(x) = −13 x^2 (5x+10)(x−3).
Set factors = 0:
x^2 = 0 → x = 0
5x+10 = 0 → x = −2
x−3 = 0 → x = 3

First Zero: (-2,0)
Second Zero: (0,0)
Third Zero: (3,0)

Step 2 — evaluate points between zeros (showing work)
h(−1) = −13(−1)^2(5(−1)+10)(−1−3) = −13·1·5·(−4) = −13·(−20) = 260
h(1) = −13(1)^2(5·1+10)(1−3) = −13·1·15·(−2) = −13·(−30) = 390

h(−1) = 260
h(1) = 390

Part 4 — end behavior
Degree: 4 (x^2·(5x+10)·(x−3) gives degree 2+1+1 = 4)
Leading coefficient: −13·5 = −65 (leading term −65x^4)

As x → −∞, h(x) → −∞, and as x → +∞, h(x) → −∞.

Explanation: The polynomial has even degree (4) so both ends point the same way; the leading coefficient is negative (−65) so both ends go down toward −∞.