Question

graph y = -2| x - 1| + 4.
what is the vertex of the graph
what are the domain range of the function
what ar the x- and y- intercept
can yuo esplain how to do it i cant seem to get it rite and i have tried alot!!!!m i got -2
please help and dont give smart rimarks plaes just help me.

Answer 1

Let's see if I can give you a hand. With absolute values, the easiest way to solve them is usually to consider when the inside is positive and when the inside is negative separately. So let's start by figuring out where those are:
x-1 = 0 when x = 1;
if x > 1, x-1 is positive, so |x-1| = x-1
if x < 1, x-1 is negative, so |x-1| = -(x-1), or 1-x

When graphing absolute value problems, this is often the easiest approach - split the function into 2 "easier" parts on either side of the point where the absolute value changes and solve it from there.
Let's do this. Then y=-2|x-1|+4 becomes:
y=-2(x-1)+4 = 6 - 2x when x >= 1
y=-2(-(x-1))+4 = 2x + 2 when x <= 1
You'll notice that at x=1, the two equations give you the same answer, y = 4

Now we can graph these two equations in their respective domains - starting at 1 and moving left (negative) along the x axis we graph y = 2x + 2 and starting at 1 and moving right (positive) along the x axis we graph y = 6 - 2x. This will give you a V-shaped graph.

The vertex of the graph is either the largest or smallest value the graph will ever reach - when you graph it you'll notice that the tip of the V, the point where the term inside the absolute value equals 0, is the vertex; It's either the greatest or smallest value the graph will ever take on, depending on whether the V is facing up or down. To solve for the location, simply take the x coordinate of the vertex, in this case 1, and plug it into either the original equation or one of the two "easier" equations we came up with and solve for y. In this case, solving any of the above equations for x = 1 gives you y = 4, so (1,4) is your vertex.

The domain is the values of x that can be plugged into the function and yield a solution. In this case, every x you plug in has a corresponding y value, so the domain of the function is all x. The range of the function is all the possible values of y you can get by plugging some value of x into the equation. You'll notice when you graph it that the function never gets bigger than 4, but can get as small as you want, so the range is y <= 4

The x and y intercepts are where the function crosses the x and y axes. The y intercept will always be found at x=0, so you can plug x=0 into your starting equation or the proper "easier" equation (in this case y=2x+2 since 0 <= 1) and solve for y. Here, y=2x+2 = 2*0+2 = 2, so your y-intercept is (0,2).

The x intercept(s) will occur anywhere where y = 0. For absolute value problems, there will either be two or 0 of them, depending on the equation. To find them, set your two "easier" equations equal to 0 and solve them for x - if you can, those points are x intercepts; if you can't, there are no x intercepts.
So for this problem, we'd do:
2x+2 = 0 -> 2x = -2 -> x = -1
6-2x = 0 -> 6=2x -> x = 3
So there are two x intercepts, at (-1, 0) and at (3,0)


I hope this helps and you follow it all; if you have any other questions, please ask!

-Tim