I'm stuck on part (c). So far I have:
At Q F(6)=-1
a|6-b|-1=-1
a|6-b|=0.
Now what?
Many thanks
+33616 You got as far as a|6 - b| = 0. This must mean that either a = 0 or b = 6.
If a = 0 then the function would reduce to f(x) = -1, a constant. Since f(x) clearly isn't a constant we can't have a = 0, so we must have b = 6.
So the function is f(x) = a|x - 6| - 1
Use the point (0, 11) to find a.
11 = a*|0 - 6| - 1
11 = a*6 - 1
12 = a*6
a = 2
Finally,
f(x) = 2|x - 6| - 1
.
+23246 I would approach this a little differently.
It is an absolute value equation, so its general form is: y - k = a|x - h|
where the vertex is located at the point (h, k) and its slope is a:
In function form, this becomes: f(x) = a|x - h| + k (notice the change-in-sign of the k-term)
For this problem, h = 6 and k = -1.
To find a, which is the slope, use the formula: (y2 - y1)/(x2 - x1).
We can use (0, 11) for (x1, y1) and (6, -1) for (x2, y2), so a = (-1 - 11)/(6 - 0) = -12/6 = -2.
However, being an absolute value graph, the slope will be negative if it opens downward, positive if it opens upward; so we have to change the -2 to a +2. (We got the negative because we used a point where the line fell downward to the vertex; if we used a point to the right of the vertex, where the line went upward, we would have gotten a +2.)
This makes our function: f(x) = 2|x - 6| - 1
Thus, a = 2 and b = 6.
Woaww! That was so confusing! You sure you aren't overcomplicating it? Isn't there a simpler and quicker way of doing it?
If a|6-b|=0, then either a=0 or |6-b|=0. Which do you think it is ?
Having decided on that, substitute the co-ordinates of P.
+128578 Let's see if this helps......."b" shows how far the basic graph of y = lxl is shifted either left or right. Since the "vertex" of the basic graph is at (0, 0), the "vertex" of this graph lies at (6, -1). Thus, the graph is shifted to the right by 6 units, so b = 6. "a" actually isn't the slope, it serves to either widen or narrow the basic graph (this one is narrower)....well anyway, we can find "a" thusly:
We have the point on the graph (0, 11)....so
y = alx - 6l - 1 and filling in for x and y, we have
11 = al0 - 6l - 1 add 1 to both sides
12 = al-6l
12 = a*6 divide by 6 on both sides
2 = a
Does that help??
+33616 You got as far as a|6 - b| = 0. This must mean that either a = 0 or b = 6.
If a = 0 then the function would reduce to f(x) = -1, a constant. Since f(x) clearly isn't a constant we can't have a = 0, so we must have b = 6.
So the function is f(x) = a|x - 6| - 1
Use the point (0, 11) to find a.
11 = a*|0 - 6| - 1
11 = a*6 - 1
12 = a*6
a = 2
Finally,
f(x) = 2|x - 6| - 1
.
