Question

complete the description of the piecewise function graphed below.
$f(x)= \

$$\begin{cases} \\quad \\text{if } -6 \\leq x \\leq -1 \\\\ \\quad \\text{if } -1 < x \\leq 2 \\\\ \\quad \\text{if } 2 < x \\leq 6 \\end{cases}$$

$
question help: video written example

Explanation:

Step1: Find the first piece ( -6 ≤ x ≤ -1 )

The line passes through (-6, 6) and (-1, -3). The slope \( m = \frac{-3 - 6}{-1 - (-6)} = \frac{-9}{5} = -\frac{9}{5} \)? Wait, no, let's check the points. Wait, when x = -6, what's y? Wait the first line: when x = -2, y = 0? Wait no, the first segment: let's take two points. Let's see, when x = -6, the point is (-6, 6)? Wait the graph: at x = -6, the line starts, and at x = -1, the point is (-1, -3). So slope is ( -3 - 6 ) / ( -1 - (-6) ) = (-9)/5? Wait no, maybe I misread. Wait, when x = -2, y = 0? Wait no, the first line: from x = -6 to x = -1. Let's take x = -6: y = 6? x = -1: y = -3. So slope \( m = \frac{-3 - 6}{-1 - (-6)} = \frac{-9}{5} \)? Wait, no, maybe the points are (-6, 6) and (-2, 0)? Wait, when x = -2, y = 0. Then x = -1, y = -3? Wait, no, let's re-examine. The first segment: from x = -6 to x = -1. Let's take two points: (-6, 6) and (-2, 0). Then slope is (0 - 6)/(-2 - (-6)) = (-6)/4 = -3/2. Then equation: y - 6 = -3/2(x + 6). So y = -3/2 x - 9 + 6 = -3/2 x - 3. Let's check x = -2: y = -3/2*(-2) -3 = 3 -3 = 0. Correct. x = -1: y = -3/2*(-1) -3 = 3/2 -3 = -3/2? Wait no, the point at x = -1 is (-1, -3). So maybe my points are wrong. Wait, the blue dot at x = -1 is (-1, -3). So let's take (-6, 6) and (-1, -3). Then slope is (-3 -6)/(-1 - (-6)) = (-9)/5 = -9/5. Then equation: y - 6 = -9/5(x + 6). So y = -9/5 x - 54/5 + 6 = -9/5 x - 54/5 + 30/5 = -9/5 x - 24/5. Wait, when x = -2: y = -9/5*(-2) -24/5 = 18/5 -24/5 = -6/5 = -1.2, but the graph at x = -2, y = 0? No, maybe I messed up. Wait, maybe the first segment is from x = -6 to x = -2? Wait, the x-axis: -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. The first line: when x = -6, y = 6; x = -2, y = 0. Then x = -1, the point is (-1, -3). Wait, no, the first segment is from x = -6 to x = -1? Wait, the domain for the first piece is -6 ≤ x ≤ -1. So let's find the equation of the line through (-6, 6) and (-1, -3). Slope m = (-3 - 6)/(-1 - (-6)) = (-9)/5 = -9/5. Then equation: y = -9/5 x + b. Plug in (-6, 6): 6 = -9/5*(-6) + b → 6 = 54/5 + b → b = 6 - 54/5 = 30/5 -54/5 = -24/5. So y = -9/5 x -24/5. Wait, but when x = -2, y = -9/5*(-2) -24/5 = 18/5 -24/5 = -6/5 = -1.2, but the graph at x = -2, y = 0? Maybe I made a mistake. Wait, maybe the first segment is from x = -6 to x = -2, with y-intercept? Wait, no, the problem says the first piece is -6 ≤ x ≤ -1. Let's check the point at x = -1: it's a blue dot, so included. So the first piece: line from (-6, 6) to (-1, -3). So equation: y = -3/2 x - 3? Wait, no, let's recalculate. Wait, if x = -6, y = 6: 6 = -3/2*(-6) -3 → 6 = 9 -3 = 6. Correct. x = -1: y = -3/2*(-1) -3 = 3/2 -3 = -3/2 = -1.5. But the point is (-1, -3). So that's wrong. Wait, maybe the slope is -3. Let's see: from x = -6 to x = -1, change in x is 5, change in y is -9 (from 6 to -3). So slope is -9/5. So equation: y = -9/5 x + c. At x = -6, y = 6: 6 = -9/5*(-6) + c → 6 = 54/5 + c → c = 6 - 54/5 = -24/5. So y = -9/5 x -24/5. Let's check x = -1: y = -9/5*(-1) -24/5 = 9/5 -24/5 = -15/5 = -3. Correct! So the first piece is \( y = -\frac{9}{5}x - \frac{24}{5} \) for \( -6 \leq x \leq -1 \).

Step2: Second piece: -1 < x ≤ 2

This is a horizontal line. The open circle at x = -1 is at y = -4? Wait, no, the open circle at x = -1 is (-1, -4), and the closed circle at x = 2 is (2, -4). So the equation is y = -4 for \( -1 < x \leq 2 \).

Step3: Third piece: 2 < x ≤ 6

This is a line from (2, -5) (open circle) to (6, -1). Wait, the open circle at x = 2 is (2, -5), and the closed circle at x = 6 is (6, -1). So slope \( m = \frac{-1 - (-5)}{6 - 2} = \frac{4}{4} = 1 \). Equation: y - (-5) = 1*(x - 2) → y + 5 = x - 2 → y = x - 7. Let's check x = 6: y = 6 -7 = -1. Correct. x = 2: y = 2 -7 = -5. Correct. So the third piece is \( y = x - 7 \) for \( 2 < x \leq 6 \).

Wait, let's verify each piece:

1. First piece: \( -6 \leq x \leq -1 \): \( y = -\frac{9}{5}x - \frac{24}{5} \). At x = -6: \( -\frac{9}{5}*(-6) - \frac{24}{5} = \frac{54}{5} - \frac{24}{5} = \frac{30}{5} = 6 \). Correct. At x = -1: \( -\frac{9}{5}*(-1) - \frac{24}{5} = \frac{9}{5} - \frac{24}{5} = -\frac{15}{5} = -3 \). Correct.

2. Second piece: \( -1 < x \leq 2 \): \( y = -4 \). The open circle at x = -1 is (-1, -4) (so not included), and closed circle at x = 2 is (2, -4) (included). Correct.

3. Third piece: \( 2 < x \leq 6 \): \( y = x - 7 \). At x = 2: \( 2 -7 = -5 \) (open circle, correct). At x = 6: \( 6 -7 = -1 \) (closed circle, correct).

So the piecewise function is:

\( f(x) = \begin{cases} -\dfrac{9}{5}x - \dfrac{24}{5} & \text{if } -6 \leq x \leq -1 \\ -4 & \text{if } -1 < x \leq 2 \\ x - 7 & \text{if } 2 < x \leq 6 \end{cases} \)

Answer:

For \( -6 \leq x \leq -1 \): \( \boldsymbol{-\dfrac{9}{5}x - \dfrac{24}{5}} \)
For \( -1 < x \leq 2 \): \( \boldsymbol{-4} \)
For \( 2 < x \leq 6 \): \( \boldsymbol{x - 7} \)