Question

1. $y = 3.2x + 9.6$ 15. the amount of fertilizer $y$ (in cups) that is needed for $x$ square feet of grass is $y = \frac{1}{4}x$. \

a. graph the equation. \
b. interpret the slope and $y$-intercept.

Explanation:

Part a: Graph the equation \( y = \frac{1}{4}x \)

Step 1: Identify the type of equation

The equation \( y = \frac{1}{4}x \) is a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y=\frac{1}{4}x \), the slope \( m=\frac{1}{4} \) and the y - intercept \( b = 0 \). This means the line passes through the origin \((0,0)\).

Step 2: Find another point on the line

We can use the slope to find another point. The slope \( m=\frac{\text{rise}}{\text{run}}=\frac{1}{4} \). Starting from the origin \((0,0)\), if we move 4 units to the right (run = 4) and 1 unit up (rise = 1), we get the point \((4,1)\).

Step 3: Plot the points and draw the line

• Plot the points \((0,0)\) and \((4,1)\) on the coordinate plane.
• Draw a straight line passing through these two points. This line represents the equation \( y=\frac{1}{4}x \).

Part b: Interpret the slope and y - intercept

Slope interpretation

The slope \( m=\frac{1}{4} \) in the equation \( y = \frac{1}{4}x \) means that for each additional square foot of grass (\( x \) increases by 1), the amount of fertilizer needed (\( y \)) increases by \( \frac{1}{4} \) cups. In other words, the rate of change of the amount of fertilizer with respect to the area of grass is \( \frac{1}{4} \) cups per square foot.

Y - intercept interpretation

The y - intercept \( b = 0 \) in the equation \( y=\frac{1}{4}x \) (since the equation is \( y=\frac{1}{4}x+0 \)) means that when the area of grass \( x = 0 \) square feet, the amount of fertilizer needed \( y = 0 \) cups. This makes sense in the context, because if there is no grass, we need no fertilizer.

Final Answers

• a. The graph is a straight line passing through \((0,0)\) and \((4,1)\) (and other points determined by the slope).
• b. Slope: For each additional square foot of grass, the fertilizer needed increases by \( \frac{1}{4} \) cups. Y - intercept: When there is 0 square feet of grass, 0 cups of fertilizer are needed.

Answer:

Part a: Graph the equation \( y = \frac{1}{4}x \)

Step 1: Identify the type of equation

The equation \( y = \frac{1}{4}x \) is a linear equation in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For \( y=\frac{1}{4}x \), the slope \( m=\frac{1}{4} \) and the y - intercept \( b = 0 \). This means the line passes through the origin \((0,0)\).

Step 2: Find another point on the line

We can use the slope to find another point. The slope \( m=\frac{\text{rise}}{\text{run}}=\frac{1}{4} \). Starting from the origin \((0,0)\), if we move 4 units to the right (run = 4) and 1 unit up (rise = 1), we get the point \((4,1)\).

Step 3: Plot the points and draw the line

• Plot the points \((0,0)\) and \((4,1)\) on the coordinate plane.
• Draw a straight line passing through these two points. This line represents the equation \( y=\frac{1}{4}x \).

Part b: Interpret the slope and y - intercept

Slope interpretation

The slope \( m=\frac{1}{4} \) in the equation \( y = \frac{1}{4}x \) means that for each additional square foot of grass (\( x \) increases by 1), the amount of fertilizer needed (\( y \)) increases by \( \frac{1}{4} \) cups. In other words, the rate of change of the amount of fertilizer with respect to the area of grass is \( \frac{1}{4} \) cups per square foot.

Y - intercept interpretation

The y - intercept \( b = 0 \) in the equation \( y=\frac{1}{4}x \) (since the equation is \( y=\frac{1}{4}x+0 \)) means that when the area of grass \( x = 0 \) square feet, the amount of fertilizer needed \( y = 0 \) cups. This makes sense in the context, because if there is no grass, we need no fertilizer.

Final Answers

• a. The graph is a straight line passing through \((0,0)\) and \((4,1)\) (and other points determined by the slope).
• b. Slope: For each additional square foot of grass, the fertilizer needed increases by \( \frac{1}{4} \) cups. Y - intercept: When there is 0 square feet of grass, 0 cups of fertilizer are needed.