So, I got this question from my teacher. I've tried solving it but to no avail. I cannot work out a method either, which can help me solve this.
So, it gives us two absolute value functions
$g(x) = 4|x-3|+3$
$f(x) = -6|x-3|+9$
There are two points, A & B as shown in the figure given below.
I don't understand, when I find the points of intersection by solving
$g(x) = f(x)$
then what are those $x$ values. Are they the coordinates of point A and B?
And what would be the next steps leading to the complete solution of this question.
I'd appreciate if anyone points out the steps or just solves it for my ease.
Best Answer
From your diagram, we can see that we must use $A,B$, and the vertex of $f(x)$ to calculate the area of our triangle. $A, B$ are the intersection points of $f(x)$ and $g(x)$.
Thus, we set $g(x) = f(x)$ to solve for our intersection points, and we get $4|x-3| +3 = -6|x-3|+9$. Simplifying, we get $10|x-3| = 6$, so $|x-3| = 0.6$. The two values that have an absolute value of $0.6$ are $0.6 $ and $-0.6$. Thus, we have $x-3 = 0.6$ or $x-3 = -0.6$. Solving for $x$, in each case, we get $x=2.4, 3.6$.
We plug these values into either $f(x)$ or $g(x)$, it doesn't matter because both functions are equal at our intersection points. So, we have $g(2.4) = 4|2.4-3| +3 = 5.4$ and $g(3.6) = 4|3.6-3|+3=5.4$. Thus, our intersection points are $(2.4,5.4)$ and $(3.6,5.4)$.
Now, we have to find the vertex of $f(x) = -6|x−3|+9$. Looking at our diagram, we see that the vertex of $f(x)$ is when it is the biggest. Since we are subtracting $-6|x−3|$ in $f(x)$, we try to make it as small as possible to make $f(x)$ as big as possible. Since absolute value is never negative, the smallest we can make it is $0$. We achieve $-6|x−3|=0$ when $x=3$. So, the vertex of $f(x)$ has $x$-coordinate $3$, and the $y$-coordinate is just $f(3) = 9$ because we made $-6|x−3|=0$. Thus, the vertex of $f(x)$ is $(3,9)$.
So, we have all the points of our triangle $(2.4,5.4), (3.6,5.4), (3,9)$. The base of our triangle is between the points $(2.4,5.4)$ and $(3.6,5.4)$ and the distance between them is $3.6-2.4 = 1.2$. So $b=1.2$. The height of our triangle is the distance from $(3,9)$ to our base. The $y$-coordinate of $(3,9)$ is $9$, and the $y$-coordinate of our base is $5.4$. The difference between the $y$ is $3.6$, so height $h = 3.6$. Use area of triangle formula $A =bh/2$ to get $A = 1.2*3.6/2 =2.16$. Thus, the area of the shaded traingle is $2.16$.