Exercise 10.1 Question 1: Determine the gradient/slope and the \( y \) - intercept of each of the following straight-line graphs: a. \( f(x)=3 x+1 \) b. \( y=x-4 \) c. \( y=-x+5 \) d. \( h(x)=\frac{1}{2} x \) e. \( y=\frac{3 x}{2}+1 \)

Function by following steps: - step0: Find the y-intercept: \(f\left(x\right)=3x+1\) - step1: Set \(x\)=0\(:\) \(f\left(0\right)=3\times 0+1\) - step2: Multiply: \(f\left(0\right)=0+1\) - step3: Remove 0: \(f\left(0\right)=1\) Analyze the y intercept of the function \( h(x)=\frac{1}{2} x \) Function by following steps: - step0: Find the y-intercept: \(h\left(x\right)=\frac{1}{2}x\) - step1: Set \(x\)=0\(:\) \(h\left(0\right)=\frac{1}{2}\times 0\) - step2: Multiply: \(h\left(0\right)=0\) Analyze the y intercept of the function \( y=x-4 \) Function by following steps: - step0: Find the y-intercept: \(y=x-4\) - step1: Set \(x\)=0\(:\) \(y=0-4\) - step2: Remove 0: \(y=-4\) Analyze the y intercept of the function \( y=-x+5 \) Function by following steps: - step0: Find the y-intercept: \(y=-x+5\) - step1: Set \(x\)=0\(:\) \(y=-0+5\) - step2: Evaluate: \(y=0+5\) - step3: Remove 0: \(y=5\) Analyze the slope of the function \( h(x)=\frac{1}{2} x \) Function by following steps: - step0: Find the slope: \(h\left(x\right)=\frac{1}{2}x\) - step1: Write the function as an equation: \(y=\frac{1}{2}x\) - step2: Identify the slope: \(m=\frac{1}{2}\) Analyze the y intercept of the function \( y=\frac{3 x}{2}+1 \) Function by following steps: - step0: Find the y-intercept: \(y=\frac{3x}{2}+1\) - step1: Set \(x\)=0\(:\) \(y=\frac{3\times 0}{2}+1\) - step2: Multiply: \(y=\frac{0}{2}+1\) - step3: Simplify: \(y=1\) Analyze the slope of the function \( y=x-4 \) Function by following steps: - step0: Find the slope: \(y=x-4\) - step1: Identify the slope: \(m=1\) Analyze the slope of the function \( y=-x+5 \) Function by following steps: - step0: Find the slope: \(y=-x+5\) - step1: Identify the slope: \(m=-1\) Analyze the slope of the function \( f(x)=3 x+1 \) Function by following steps: - step0: Find the slope: \(f\left(x\right)=3x+1\) - step1: Write the function as an equation: \(y=3x+1\) - step2: Identify the slope: \(m=3\) Let's determine the gradient (slope) and the \( y \)-intercept for each of the given straight-line graphs step by step. ### a. \( f(x) = 3x + 1 \) - **Slope (m)**: The coefficient of \( x \) is \( 3 \), so \( m = 3 \). - **Y-intercept**: When \( x = 0 \), \( f(0) = 3(0) + 1 = 1 \). Thus, the \( y \)-intercept is \( 1 \). ### b. \( y = x - 4 \) - **Slope (m)**: The coefficient of \( x \) is \( 1 \), so \( m = 1 \). - **Y-intercept**: When \( x = 0 \), \( y = 0 - 4 = -4 \). Thus, the \( y \)-intercept is \( -4 \). ### c. \( y = -x + 5 \) - **Slope (m)**: The coefficient of \( x \) is \( -1 \), so \( m = -1 \). - **Y-intercept**: When \( x = 0 \), \( y = -0 + 5 = 5 \). Thus, the \( y \)-intercept is \( 5 \). ### d. \( h(x) = \frac{1}{2} x \) - **Slope (m)**: The coefficient of \( x \) is \( \frac{1}{2} \), so \( m = \frac{1}{2} \). - **Y-intercept**: When \( x = 0 \), \( h(0) = \frac{1}{2}(0) = 0 \). Thus, the \( y \)-intercept is \( 0 \). ### e. \( y = \frac{3}{2} x + 1 \) - **Slope (m)**: The coefficient of \( x \) is \( \frac{3}{2} \), so \( m = \frac{3}{2} \). - **Y-intercept**: When \( x = 0 \), \( y = \frac{3}{2}(0) + 1 = 1 \). Thus, the \( y \)-intercept is \( 1 \). ### Summary of Results: - **a.** Slope: \( 3 \), Y-intercept: \( 1 \) - **b.** Slope: \( 1 \), Y-intercept: \( -4 \) - **c.** Slope: \( -1 \), Y-intercept: \( 5 \) - **d.** Slope: \( \frac{1}{2} \), Y-intercept: \( 0 \) - **e.** Slope: \( \frac{3}{2} \), Y-intercept: \( 1 \)