Question

modeling data with functions: mastery test
1
select the correct answer.
george is drafting his dissertation paper for his doctoral program. the graph below shows a linear model for the data recorded on the number of pages of the dissertation written as a function of the number of hours george spends on the draft each week.
graph of a scatter plot with a linear model line. the x-axis is labeled no. of hours per week from 0 to 20, and the y-axis is labeled no. of pages written from 0 to 20. the line passes through (0, 4) and other points.
select the equation for the linear model shown
a. $y = \frac{1}{2}x + 4$
b. $y = \frac{2}{3}x + 4$
c. $y = \frac{3}{2}x + 4$
d. $y = 2x + 4$
undo next

Explanation:

Step1: Identify the y-intercept

The linear equation is in the form \( y = mx + b \), where \( b \) is the y-intercept. From the graph, when \( x = 0 \), \( y = 4 \), so \( b = 4 \).

Step2: Calculate the slope \( m \)

Choose two points on the line, e.g., \( (0, 4) \) and \( (4, 6) \). The slope formula is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Substituting values: \( m=\frac{6 - 4}{4 - 0}=\frac{2}{4}=\frac{1}{2} \). Wait, no, let's check another point. Wait, maybe \( (0,4) \) and \( (6,7) \)? Wait, no, let's take \( (0,4) \) and \( (4,6) \): \( m=\frac{6 - 4}{4 - 0}=\frac{2}{4}=\frac{1}{2} \)? Wait, no, the options have \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{2} \), \( 2 \). Wait, maybe I made a mistake. Wait, let's take \( (0,4) \) and \( (2,5) \): \( m=\frac{5 - 4}{2 - 0}=\frac{1}{2} \). Wait, but let's check the line. Wait, the line passes through \( (0,4) \) and when \( x = 4 \), \( y = 6 \), so slope is \( \frac{6 - 4}{4 - 0}=\frac{2}{4}=\frac{1}{2} \). Wait, but option A is \( y=\frac{1}{2}x + 4 \). Wait, but let's check another point. Wait, when \( x = 4 \), \( y = 6 \): \( \frac{1}{2}(4)+4 = 2 + 4 = 6 \), which matches. When \( x = 8 \), \( \frac{1}{2}(8)+4 = 4 + 4 = 8 \), which also matches the line. So the slope is \( \frac{1}{2} \) and y-intercept is 4, so the equation is \( y=\frac{1}{2}x + 4 \). Wait, but wait, maybe I misread the graph. Wait, the points: when x=0, y=4; x=2, y=5; x=4, y=6; x=6, y=7; x=8, y=8; x=10, y=9; x=12, y=10; x=14, y=11; x=16, y=12; x=18, y=13? Wait, no, the line in the graph: the linear model. Wait, the red dots are the data, the line is the model. So the line passes through (0,4) and (4,6), so slope 1/2. So the equation is \( y=\frac{1}{2}x + 4 \), which is option A. Wait, but let's check option A: \( y=\frac{1}{2}x + 4 \). When x=4, y= (1/2)*4 +4=2+4=6, which matches. When x=8, y=4 +4=8, which matches. So that's correct.

Answer:

A. \( y = \frac{1}{2}x + 4 \)