Answer:
Explanation:
This is a piecewise function, and it seems to consist of two linear segments.
For the first segment (from the leftmost point to the vertex):
The line starts at the point (-3, -3) and ends at the vertex (4, 5).
The slope, m, of this segment is given by:
�
=
change in
�
change in
�
=
5
−
(
−
3
)
4
−
(
−
3
)
=
8
7
m=
change in x
change in y
=
4−(−3)
5−(−3)
=
7
8
So the equation of the line in point-slope form is:
�
−
(
−
3
)
=
8
7
(
�
−
(
−
3
)
)
y−(−3)=
7
8
(x−(−3))
�
+
3
=
8
7
(
�
+
3
)
y+3=
7
8
(x+3)
For the second segment (from the vertex to the rightmost point):
The line starts at the vertex (4, 5) and continues downwards with a slope of -1 (since it's a 45° line declining).
So the equation in point-slope form is:
�
−
5
=
−
1
(
�
−
4
)
y−5=−1(x−4)
�
−
5
=
−
�
+
4
y−5=−x+4
Combining these, the piecewise function is:
�
(
�
)
=
{
8
7
(
�
+
3
)
−
3
for
�
≤
4
−
�
+
9
for
�
>
4
f(x)={
7
8
(x+3)−3
−x+9
for x≤4
for x>4
Intervals of Increase, Decrease, or Constant:
Increasing:
�
≤
4
x≤4
Decreasing:
�
>
4
x>4
Intercepts:
x-intercept: Where y = 0. For the first segment,
0
=
8
7
(
�
+
3
)
−
3
0=
7
8
(x+3)−3
Solving for x, you get x is approximately -0.375 or -3/8.
y-intercept: Where x = 0. For the first segment,
�
=
8
7
(
0
+
3
)
−
3
y=
7
8
(0+3)−3
y is approximately 1.14 or 8/7.
Asymptotes:
There are no asymptotes for linear functions.
End Behavior:
As
�
→
−
∞
x→−∞,
�
(
�
)
→
−
∞
f(x)→−∞
As
�
→
∞
x→∞,
�
(
�
)
→
−
∞
f(x)→−∞