Question

Is this the correct graph for the piece wide function

2x + 6 if x < 1
f (x) = { -1/4x - 3 if 1 ≤ x < 6?
4 + 4x if x ≥ 6

Answer 1

To determine if the given graph represents the provided piecewise function correctly, let's analyze each piece of the function:

1. For \( x < 1 \):
- The function is defined as \( f(x) = 2x + 6 \).

2. For \( 1 \leq x < 6 \):
- The function is defined as \( f(x) = -\frac{1}{4}x - 3 \).

3. For \( x \geq 6 \):
- The function is defined as \( f(x) = 4 + 4x \).

To graph this piecewise function correctly, you would follow these steps:

1. **For \( x < 1 \)**:
- Plot the equation \( y = 2x + 6 \). This line has a positive slope and intersects the y-axis at \( (0, 6) \).
- This part of the graph should continue to the left of \( x = 1 \) but not include the point at \( x = 1 \) (open circle).

2. **For \( 1 \leq x < 6 \)**:
- Start at \( x = 1 \) where \( f(1) = -\frac{1}{4}(1) - 3 = -\frac{1}{4} - 3 = -\frac{13}{4} \) which is a closed point (solid circle) because \( x = 1 \) is included.
- Draw the line segment from \( (1, -\frac{13}{4}) \) to \( (6, -\frac{1}{4}(6) - 3 = -\frac{15}{4}) \). This segment does not include the endpoint \( x = 6 \) (open circle).

3. **For \( x \geq 6 \)**:
- Start at \( x = 6 \) which will equal \( f(6) = 4 + 4(6) = 28 \). This should be a closed point.
- The function is \( f(x) = 4 + 4x \), which is a linear function with a slope of 4, so the graph will steeply rise to the right.

Make sure the graph correctly reflects these behaviors at the boundaries. The transition points, specifically \( x=1 \) and \( x=6 \), are crucial to portray accurately, with open circles for exclusions and solid for inclusions.

If your graph accurately reflects these instructions, then it is correct. If you have a specific graph in mind or want to visually check it, you may post an image or describe it for further verification.