Question

owen and rose each thought of a linear function. owens function is $y = \frac{3}{4}x + 1$, and roses function is shown in the graph. which statement is true? a owens function has both a greater slope and a greater y-intercept. b roses function has both a greater slope and a greater y-intercept. c roses function has a greater slope, and owens function has a greater y-intercept. d owens function has a greater slope, and roses function has a greater y-intercept.

Explanation:

Step1: Recall linear function form

The slope-intercept form of a linear function is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Owen's function is \( y=\frac{3}{4}x + 1 \), so his slope \( m_O=\frac{3}{4}=0.75 \) and y-intercept \( b_O = 1 \).

Step2: Determine Rose's function parameters

For a linear graph, we can find two points to calculate slope. Let's assume from the graph (common for such problems) Rose's line passes through (0, 2) (y-intercept, since x=0) and (4, 4). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Using (0,2) and (4,4): \( m_R=\frac{4 - 2}{4 - 0}=\frac{2}{4}=0.5 \). Her y-intercept \( b_R = 2 \) (from x=0, y=2).

Step3: Compare slopes and y-intercepts

Slope comparison: \( m_O = 0.75 \), \( m_R=0.5 \), so Owen's slope is greater. Y-intercept comparison: \( b_O = 1 \), \( b_R = 2 \), so Rose's y-intercept is greater.

Answer:

D. Owen’s function has a greater slope, and Rose’s function has a greater y-intercept.