Certainly! To sketch the graph of the given piecewise function:
[tex]\[ f(x) = \begin{cases}
(x + 4) & \text{if } x < -4 \\
-6 & \text{if } -4 \leq x < 4 \\
(x - 5)^2 & \text{if } x > 4
\end{cases} \][/tex]
we will consider each piece of the function separately and then combine the results to form the complete graph.
### Step-by-Step Solution
1. For [tex]\( x < -4 \)[/tex]:
[tex]\[ f(x) = x + 4 \][/tex]
This is a linear function with a slope of 1 and a y-intercept of 4.
- When [tex]\( x = -5 \)[/tex], [tex]\( f(-5) = -5 + 4 = -1 \)[/tex].
- When [tex]\( x = -6 \)[/tex], [tex]\( f(-6) = -6 + 4 = -2 \)[/tex].
Plot a few points and draw a line extending to the left.
2. For [tex]\( -4 \leq x < 4 \)[/tex]:
[tex]\[ f(x) = -6 \][/tex]
This is a constant function.
[tex]\[ y = -6 \][/tex]
This will be a horizontal line extending from [tex]\( x = -4 \)[/tex] to [tex]\( x = 4 \)[/tex].
- The points [tex]\( (-4, -6) \)[/tex] and [tex]\( (4, -6) \)[/tex] are included in this interval.
3. For [tex]\( x > 4 \)[/tex]:
[tex]\[ f(x) = (x - 5)^2 \][/tex]
This is a quadratic function that opens upwards and is shifted 5 units to the right.
- When [tex]\( x = 5 \)[/tex], [tex]\( f(5) = (5 - 5)^2 = 0 \)[/tex].
- When [tex]\( x = 6 \)[/tex], [tex]\( f(6) = (6 - 5)^2 = 1 \)[/tex].
- When [tex]\( x = 7 \)[/tex], [tex]\( f(7) = (7 - 5)^2 = 4 \)[/tex].
Plot a few points and draw the parabola extending to the right.
### Combined Graph
Now, combine all the individual pieces:
- In the interval [tex]\( x < -4 \)[/tex]: Start from the left extending linearly until -4.
- In the interval [tex]\( -4 \leq x < 4 \)[/tex]: Draw a horizontal line at [tex]\( y = -6 \)[/tex].
- In the interval [tex]\( x > 4 \)[/tex]: Start from [tex]\( (5, 0) \)[/tex] and draw the upward-opening parabola for [tex]\( x > 4 \)[/tex].
### Sketch
1. Plot the points as marked from the evaluated values.
2. Draw a straight line from the left until the point [tex]\( (-4, 0) \)[/tex] for [tex]\( f(x) = x + 4 \)[/tex].
3. Draw a horizontal line from [tex]\( (-4, -6) \)[/tex] to [tex]\( (4, -6) \)[/tex].
4. Plot the points starting from [tex]\( (5, 0) \)[/tex] and extending as [tex]\( f(x) = (x - 5)^2 \)[/tex].
### Important Points
- The function jumps from [tex]\( f(-4) = -6 \)[/tex] to [tex]\( f(4) = -6 \)[/tex], showing a horizontal line.
- At [tex]\( x = 5 \)[/tex], the function starts from 0 and curves upwards as a parabola.
This description should allow you to sketch the full graph of the piecewise function accurately.
