Function Machines Activity from CPM | Math = Love - Free Printable

The image shows four "machines" with mathematical functions written on them. Each machine takes an input \( x \) and produces an output \( y \) based on the given function. The task is to understand how each machine works by analyzing the functions provided.

Step-by-Step Explanation:

#### Machine 1: \( y = -2x + 34 \)

- Function: This is a linear function in the form \( y = mx + b \), where:
- \( m = -2 \) (slope)
- \( b = 34 \) (y-intercept)

- Behavior:
- For every increase of 1 in \( x \), \( y \) decreases by 2.
- When \( x = 0 \), \( y = 34 \).

- Example:
- If \( x = 5 \):
\[
y = -2(5) + 34 = -10 + 34 = 24
\]
- If \( x = -3 \):
\[
y = -2(-3) + 34 = 6 + 34 = 40
\]

#### Machine 2: \( y = -\frac{x}{3} - 10 \)

- Function: This is also a linear function, but with a fractional slope.
- \( m = -\frac{1}{3} \) (slope)
- \( b = -10 \) (y-intercept)

- Behavior:
- For every increase of 3 in \( x \), \( y \) decreases by 1.
- When \( x = 0 \), \( y = -10 \).

- Example:
- If \( x = 6 \):
\[
y = -\frac{6}{3} - 10 = -2 - 10 = -12
\]
- If \( x = -9 \):
\[
y = -\frac{-9}{3} - 10 = 3 - 10 = -7
\]

#### Machine 3: \( y = -|3x| \)

- Function: This involves the absolute value function and a negative sign.
- The expression \( |3x| \) means the absolute value of \( 3x \), which is always non-negative.
- The negative sign in front of the absolute value makes the output non-positive.

- Behavior:
- For any positive or negative \( x \), \( |3x| \) will be positive, and the negative sign will make \( y \) negative.
- When \( x = 0 \), \( y = 0 \).

- Example:
- If \( x = 2 \):
\[
y = -|3(2)| = -|6| = -6
\]
- If \( x = -4 \):
\[
y = -|3(-4)| = -|-12| = -12
\]

#### Machine 4: \( y = (x - 2)^2 \)

- Function: This is a quadratic function in vertex form \( y = (x - h)^2 + k \), where:
- The vertex is at \( (h, k) = (2, 0) \).
- The parabola opens upwards because the coefficient of \( (x - 2)^2 \) is positive.

- Behavior:
- The minimum value of \( y \) occurs at \( x = 2 \), where \( y = 0 \).
- As \( x \) moves away from 2 in either direction, \( y \) increases.

- Example:
- If \( x = 2 \):
\[
y = (2 - 2)^2 = 0^2 = 0
\]
- If \( x = 4 \):
\[
y = (4 - 2)^2 = 2^2 = 4
\]
- If \( x = 0 \):
\[
y = (0 - 2)^2 = (-2)^2 = 4
\]

Final Answer:

Each machine transforms the input \( x \) into the output \( y \) according to the specified function. The solutions for the machines are summarized as follows:

1. Machine 1: \( y = -2x + 34 \)
2. Machine 2: \( y = -\frac{x}{3} - 10 \)
3. Machine 3: \( y = -|3x| \)
4. Machine 4: \( y = (x - 2)^2 \)

\[
\boxed{
\begin{aligned}
1. & \quad y = -2x + 34 \\
2. & \quad y = -\frac{x}{3} - 10 \\
3. & \quad y = -|3x| \\
4. & \quad y = (x - 2)^2
\end{aligned}
}
\]