Question

PLZ HELP ME!!!
Use the function f(x) = 2x2 − x − 10 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)

Answer 1

A: So there a few techniques you can utilize to do this. It was only recently where I learned the "slide and divide method". Essentially, you want to make sure that in your quadratic formula:
\[ax ^{2}+bx+c\]
You want a to equal 1. Since your coefficient is equal to 2, we'll simply divide a by 2 and multiply our c value by 2. Now when you factor that out, you will have:
\[(x-5)(x+4)\]
Now remember this isn't our completely factored form because that is not equal to the original equation given. Because we divided our a value by 2, we divide our integer values by 2 as well.
\[(x-\frac{ 5 }{ 2 })(x+\frac{ 4 }{ 2 })\]
Now take notice how the second factor can be simplified into (x-2). We perform this operation. All that is left is to fix up the first factor. Since we must have a 2x squared value in our quadratic equation, we multiply each value within the binomial of (x-5/2) by 2 to get
\[(2x-5)\]
I hope this helps for when you factor binomials in the future.

B: X-intercepts are also known as zeroes. Or to be more specific, x-intercepts are x values where y=0. Now because we found the factors of our quadratic equation, we can use it to easily figure out what x value will make the function return a value of 0. As reference, here are our binomials.
\[(2x-5)(x+2)\]
All that must be done is isolate each factor and set them each to zero, like so:
\[2x-5=0\]
\[x-2=0\]
Because the factors are multiplicative, if any of the x values in any binomial equal 0, then the whole function equals 0. Isolate and solve for x.

C: This is a pretty easy part, conceptually speaking. For this one, we take notice that this equation is a quadratic function. Because our ax squared value has the most influence in terms of power on the function, we only pay attention to this value. A rule of thumb is that if ax squared is positive, then f(x) will open upwards indefinitely. Likewise, if ax squared is negative, then f(x) will open downwards indefinitely. If we were to put this in mathematical terms, we could say
\[\lim_{x \rightarrow \infty} ax ^{2}+bx+c \rightarrow \ \]
\[\lim_{x \rightarrow -\infty} ax ^{2}+bx+c \rightarrow \ \]
(This part is intentionally left blank. This is for you to find.)
Just remember that if a quadratic function opens upwards, it will have a minimum value. Beyond that minimum point will always be increasing values. Likewise, if a quadratic function opens downwards, it will have a maximum value of f(x). Beyond that maximum point will always be decreasing values of f(x).

D: So for this part, I would most definitely use the vertex formula. It states that in order to find the minimum or maximum value of the graph, use the following:
\[\frac{ -b }{ 2a }\]
Where a is the coefficient of the x squared term and b is the second term of the quadratic function. By finding this vertex, you find out what x value holds said min or max value. By finding x, you can substitute your x value into f(x) and find out what the max or min value is. Then I would begin by marking the x values where y=0, which were found in part b. From this point on, if you really wanted to, you can plug in x values in between the vertex and the zeroes to find the other points of reference. Because a quadratic function is always symmetrical, you can simply just mark that same point, but on the other side of the vertex. Then I would draw the curve to the best of my ability, just enough to show what the end behavior is.

Answer 2

THANK YOU SO MUCH :D

Answer 3

Just realized there were a few mistakes I made, the x-2 I stated in part b should be (x+2), and the same with part a. Sorry about that.

Answer 4

can i have the answers?

Answer 5

I'm afraid I cannot give the answers, but I am allowed to guide you to your answers. Tell me what parts you're having trouble with and explain what your thought process is. If it has something to do with how I explained the problems, then please shoot questions relating to where the knowledge gap is.