D.E.V.

By Hilary5th hour

Developing yourexpert voice

Problem 1

Let’s start with an easier problem to do. In this problem, I’m going to give you an f(x) function, h(x) function, and a g(x) function and we will walk through the indicated operation. After that we will find the domain and range of the final function

F(x)= 4x²+16x G(x)= x-6 H(x)= x+2 Find H(G(F(x)))

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First find G(F(x)) by substituting the x in the G function for the F function› G(F(x))= (4x²+16x)-6

Now put this in the H function and simplify› H(G(F(x)))= 4x²+16x-4 (-6+2= -4)

4› H(G(F(x)))= x²+4x-1

Finding domain and rangeH(G(F(x)))= x²+4x-1

For quadratics, domain is all real numbers!› Domain: (-∞,∞)

With a positive a value (number in front), the range will be K (last number) to infinity› Range: [-1, ∞)

Remember to include the -1 so use a bracket around the number

Just like that we are done with the first problem.

Problem 2

In this problem I am going to give an equation in standard form. From there we will convert to factored form and vertex form. That is not all folks. We are going to find the vertex, x-intercepts and y-intercepts as well. Finally we will graph it all. Let’s get cracking.

F(x)= x²+4x-32 ← standard form Simply factor this into 2 parentheses like

normal to get factored form› F(x)= (x+8)(x-4) ← factored form

To find x- intercepts use factored form› Remember intercepts are when the

parentheses equal 0 X=-8 x=4 ← x-intercepts

To find y-intercepts use standard form› Use the c value (last number) to find intercept

(0, -32) ← y-intercept

F(x)= x²+4x-32

To find vertex form we need to complete the square using standard form

To find the vertex, use vertex form for obvious reasons› Use the opposite of the number in parentheses

(make equal to 0) and number on outside (-2, -36) ← vertex

Graph

Plot all points before you make graph

Problem 3

Before we start this, it’s okay if you take a quick breather. Math can be a lot to handle sometimes. I’m not judging.

Let’s move on to learn a bit about rationals. I’m going to give a function and together we will find x and y-intercepts, holes, vertical and horizontal asymptotes. Not too shabby if I do say so myself.

F(x)= x²+4x-213x²+9x-84

Factor both the numerator and denominator. Make sure to look for greatest common factor.

The factored function should look like› (x+7)(x-3) x+7 can be divided out

(hole!)3(x+7)(x-4)

The function is now› (x-3) with a hole at x= -7 (makes it 0!)

3(x-4)

(x-3) x²+4x-21 3(x-4) 3x²+9x-84

To find x- intercepts look to the numerator in factored form. We need to find the number that will make the numerator 0.› In this case the x-int. is (3,0)

To find the y- intercept we look at standard form. We plug 0 in for x and see what is left over which in this case is -21/-84.› After simplifying the y-intercept is (0, ¼)

(x-3) x²+4x-21 3(x-4) 3x²+9x-84

To find the horizontal asymptote, we divide the leading coefficients. I’ve made it easier by having the highest power in the num. and den. the same.› The H.A. is where y= 1/3

To find the vertical asymptote, we look where the denominator will equal 0 using factored form.› The V.A. is where x = -4

Graph

Problem 4

We are almost done figuring out problems together. Let’s cruise through this last problem and consider ourselves accomplished people. This last one is meant to be a challenge so don’t get flustered, we will get through it. I’m going to give a function and we will find x-intercepts, graph and find domain. Let’s do this.

F(x)=√x⁴+9x³+11x²+9x-180, x= -5 I gave a solution so we will use it to

long divide› Remember to use the rule: divide,

distribute subtract, drop. Set it up like this: (leave space on top)

Here is what the long division should look like

Almost done!

We have one solution (x= -5) and x³=4x²-9x-36. Luckily this one groups! Here is how to do it.

Graph Our 4 solutions: x=3, x= -3, x= -4, x= -5 We have a positive a value and an even

degree (number of solutions) so graph will look like a w.

Domain has to be where y is positive› Domain: (-∞, -5]U[-4, -3]U[3,∞] (include

intercepts with brackets) ↑ means union

Thanks for looking through my D.E.V. project. I hope that it was at least a little bit helpful for you. This project may have been hard but it was worth it.“Tell me and I forget. Teach me and I remember. Involve me

and I learn.”-Benjamin Franklin

Swag

Brief Reflection

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