100 POINTS Use the function f(x) = 5x2 + 2x − 3 to answer the questions. Part A: Completely factor f(x). (2 points) Part B: What
are the x-intercepts of the graph of f(x)? Show your work. (2 points) Part C: Describe the end behavior of the graph of f(x). Explain. (2 points) Part D: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part B and Part C to draw the graph. (4 points)
1 answer:
Answer:
see below
Step-by-step explanation:
f(x) = 5x^2 + 2x − 3
(5x-3) ( x+1)
To find the x intercepts, set this equal to zero
0= (5x-3) ( x+1)
Using the zero product property
5x-3 =0 x+1 =0
5x=3 x =-1
x = 3/5 x = -1 are the zeros
End behavior
x = -infinity
5 ( -infinity )^2
5 (infinity) = infinity
as x goes to negative infinity, f(x) goes to infinity
x = infinity
5 ( infinity )^2
5 (infinity) = infinity
as x goes to infinity, f(x) goes to infinity
To plot the function, we need to plot the zeros ( or x intercepts) and the end behavior as well as the vertex to plot the graph
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There are a variety of methods of performing this subtraction, and corresponding different methods of regrouping.
If you're taught (as I was) to do the subtraction right-to-left, then the first regrouping you need to do is when you try to subtract 500 from 100. You must regroup the 8 thousands and 1 hundred to 7 thousands and 11 hundreds. Then, when you subtract 5 hundreds, you end with 7 thousands and 6 hundreds.
The next regrouping you need to do is when you try to subtract 6 ten-thousands from 5 ten-thousands. You must regroup the 2 hundred-thousands and 5 ten-thousands to 1 hundred-thousand and 15 ten-thousands. Then, when you subtract 6 ten-thousands, you end with 1 hundred-thousand and 9 ten-thousands.
The end result is
... 258,197 - 64,500 = 193,697
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If you use an abacus or soroban or similar tool to help you keep track of the numbers, you were likely taught to do the subtraction left-to-right. In this case, the first regrouping comes when you want to subtract 6 ten-thousands. Practitioners of this method know that -6 = -10 +4, so the number represented on the tool becomes (2-1) hundred-thousands and (5+4) ten-thousands plus the rest of the initial number, or 198,197 after subtracting the 6 ten-thousands.
The subtraction proceeds until you find you need to subtract 500 from 100. At this point, the tool is representing the partial result as 194,197. Again, if you practice this method, you know that -5 = -10 +5, so you reduce the thousands digit by 1 (to 3) and add 5 to the hundreds digit to get 193,697.
_____
An attempt is made to show the regroupings in the attachment. In each case there are two of them. However, working left-to-right, the result of the first subtraction of 6 ten-thousands is 19 ten-thousands, so you never actually write down anything else. Of course, if you're using an abacus or soroban, you don't write down anything—you simply change the position of the beads on the tool.
