Parallel lines in the coordinates planes

Question

The graph of g(x) goes through the point (3,1) and is parallel to h(x) = -4x + 5. Determine the symbolic representation of g(x). Express your answer in slope-intercept form.

Answer 1

So we want g(x) to be parallel to h(x), and we want to express our answer g(x) in slope-intercept form.

Slope intercept form in general looks like f(x)=mx+b or y=mx+b, where m and b are two constants. Here, m would be the slope and b would be considered the intercept.

The given h(x) from the problem is in slope-intercept form already: h(x)= -4x+5.

So here for h(x), -4 is the slope of the line, and 5 is the intercept. Specifically, this 5 is telling you the "y-intercept", where the graph crosses the y-axis with an x value of 0.

Right now we are more interested in the slope of this line, -4.

Parallel lines have the same slope, that is part of the definition of parallel, they have the same slope and they do not intersect.

So if we are trying to find g(x) and have it be parallel to h(x), we know its going to have the same slope as h(x), so its slope will be -4 as well.

So for our slope-intercept form for g(x): g(x)=mx+b or y=mx+b, we already know one of those two constants, the one corresponding to slope: m

I will fill that in: g(x)=-4x+b

Now we just have to find b. To do that we can use the last piece of information we have from the question: that g(x) goes through the point (3,1).

We can plug in this point (3,1) into our y=mx+b form for the x and y values (x,y).

So we can put in x=3 and y=1 into our equation g(x)=-4x+b, since we know that g(x) goes through that point (3,1). That looks like:

1=-4(3)+b, and we want to solve for b, the last constant we need for the g(x) equation

1=-4(3)+b

1=-12+b

13=b

So lets put it all together to get our equation for g(x) in slope-intercept form:

g(x) = -4x + 13

Answer 2

You have a symbolic representation of h(x), which is a line. Now you are asked to come up with a, more or less, similar symbolic representation of g(x), where x is the input or domain and g(x) is the output or range.

A key piece of information is that the two lines, g(x) and h(x), are PARALLEL. Anytime that you hear that lines are parallel, a little bell should go off in your head, and you should hear the words, "Parallel lines have the same slope."

Looking at h(x), what is the slope. Well, the line is written in slope-intercept form, so you can read off the slope. It is the coefficient of the x term. h(x) = –4x + 5. The coefficient of x is –4; that is the SLOPE.

So, we start with the template: g(x) = ?x + ? However, now we can fill in the slope; that is, we can fill in the coefficient of x. So, now we have: g(x) = –4x + ?

The other clue that we have about g(x) is that point (3,1) is on the line. What this means is that if you use x=3, then g(3) = 1. So, we go back to our partway completed equation (or symbolic representation) for g(x) and fill in this new information. We get: g(3) = 1= –4(3) + ?

You may want to use a symbol other than a question mark, but I'm going to stick with it and solve for ?

1 = –4(3) + ?

1 = –12 + ?

1 + 12 = –12 + 12 + ?

13 = ?

Now, I go back and fill in 13 for the last question mark. I get: g(x) = –4x + 13

Lots of students get mixed up with function notation. You can look at the g function as a black box. When you stick x into it, you get –4x + 13 out of it. If you stick a lot of different x's in it, you will get a bunch of different outputs. The pairs of inputs and their corresponding outputs are ordered pair that can be plotted on a coordinate plane to form a line in this case.