I dont have options but i can sayEach of these questions is asking you to figure out how g(x) would be graphed based on some other function, f(x), except we're gonna make some changes to f(x) first. For example, let's look at part (a). You're told that g(x) = 2f(x)+3. 1. Whatever f(x) is, 2f(x) would take all of the y values and double them.2. Whatever that turns out to be, "+3" would add 3 to every y value. So, to translate (1, -2), first we double the y value...so that'd be -2 * 2, or -4. Then we add 3, which gives 1. So, the point (1, -2) becomes (1, 1). [EDIT: Um, no, -4 + 1 isn't 1. It's -1. My bad! MBW] Part (b) is a little bit trickier: 1. Whatever f(x) is, by adding 1 to x, it actually shifts the graph left by 1, even though it sorta feels like you should be shifting it 1 to the right. Let's not get too far into the details here, so for the moment, f(x+1) means "shift the graph of f to the left by 1." In other words, subtract 1 from x to get the new point. 2. Whatever f(x+1) is, then the "-3" would subtract 3 to every y value. So, if (1, -2) was on the original graph, then f(x+1) would be shifted to the left by 1...so that's (0, -2). And then, we subtract 3 from y, so that'd be (0, -5). Basically, anything inside the parentheses, like f(x+1), messes with the x coordinate of the point, and anything outside the parentheses, like -f(x) or f(x) + 3, messes with the y coordinate. For the last two, I'll give you a few hints and see if you can take it from there. For part (c): f(2x), even though you think it might double the value of x, actually divides the value of x by 2.For part (d): the trick here is to ignore the "-x" until the last step. Deal with the f(x-1) first (which shifts the x coordinate...which way?), then the negative outside of f(x) (which flips the sign of the y coordinate), then the +3 outside of f(x)...and then, at the end, the "-x" would flip the sign of the x coordinate. I hope this helps point you in the right direction!
Hi! This answer may seem too big, but I suggest taking the time to read it as it will help you a lot in the future.
<h3>Step 1:</h3>
Let's start by multiplying -(m - 6).
Essentially, you are turning the expression (m - 6) negative or multiplying the expression (m - 6) by -1
This uses the distributive property of multiplication (distributing the outside factor to all the terms inside the parenthesis. You'll see how it works soon.)
Let's solve:
-(m - 6)
-1(m - 6)
-1(m) - 1(-6)
<h3>PAUSE</h3>
Here are a couple of tips:
<h2>Tips:</h2>
A positive number times a negative number is always a negative number
A positive number times a positive number is always a positive number
A negative number times a negative number is always a positive number
A number times 1 is the same number
A number times -1 is the opposite of that number
So basically, if you are multiplying terms with the same sign (positive or negative), it is always positive. Numbers with opposite signs are always negative.
<h3>CONTINUE</h3>
Using the information we have now, we can multiply the terms
-1(m) - 1(-6)
-m + 6
6 - m
The left-hand side of the equation becomes 6 - m
<h3>Step 2:</h3>
Now, let's solve for m.
Solve for m:
<h2>Tips:</h2>
When dividing a positive number by a negative number, the answer is always negative
When dividing a negative number by a positive number, the answer is always negative
When dividing a positive number by a positive. number, the answer is always positive
When dividing a negative number by a negative number, the answer is always positive.
When dividing a number by -1, it is the opposite of that number
When dividing a number by 1, it is the same number.
The same rules apply for division as they do to multiplication as well.
What you do to one side of the equation should happen to the other side.
Step 1: 6 - m = 3m + 14
Step 2: 6 = 4m + 14
Step 3: -8 = 4m
Step 4: 4m = -8
Step 5: m = -2
Analyze:
When solving for variables, you want all the terms with that variable to one side.
In the first step, we are subtracting m. To counter that operation, we add m to both sides.
Similarly in step 2, we are adding 14, so to counter that, we subtract 14.
In steps 3 and 4, we are multiplying m by 4. To counter that, we divide by 4