Which equations give the correct quiz score when $x = 0$? Use that to cross out any equations that cannot fit the data well.

Compare the quiz scores for students who studied 0 hours and 4 hours. How many points higher is the score, and over how many hours did that change occur?

Take your estimate of how many points the score increases per hour and compare it to the $x$-coefficients (the slopes) in the remaining equations.

Create a table and enter the $x$-values 0, 1, 2, 3, 4 in the first column and the corresponding $y$-values 65, 70, 74, 80, 83 in the second column. This will plot the five data points as a scatterplot.

On separate lines, type each option exactly: y = 3x + 65, y = 4.5x + 65, y = 5x + 60, and y = 6x + 55. Desmos will draw four lines over your scatterplot.

Look for the line that goes through the point corresponding to 0 hours (the first data point) and stays closest to the rest of the plotted points overall. The equation of that line is the best model among the choices.

We are looking for a linear equation $y = mx + b$ whose graph (a straight line) fits the pattern of the data points in the table. The line does not need to go through every point perfectly, but it should match the overall trend: how $y$ changes as $x$ increases.

From the table, when $x = 0$ hours, the quiz score is $y = 65$. In a line $y = mx + b$, the value of $b$ is the $y$-intercept, which is the $y$-value when $x = 0$.

So here the model should have $b = 65$ because $y$ is 65 when $x = 0$. That means any good equation should give 65 when $x = 0$, so you can eliminate any option whose constant term is not 65.

Check each answer’s constant term (the number added at the end):

• $y = 3x + 65$ has $b = 65$.
• $y = 4.5x + 65$ has $b = 65$.
• $y = 5x + 60$ has $b = 60$.
• $y = 6x + 55$ has $b = 55$.

Since the data show $y = 65$ when $x = 0$, equations with $b = 60$ or $b = 55$ do not match that point well, so you can discard $y = 5x + 60$ and $y = 6x + 55$. You are left comparing only the two equations with $b = 65$.

Now compare the slopes of the remaining equations: 3 and 4.5. The slope tells you how many points the quiz score increases per extra hour of studying.

Use two points that are far apart in the table to estimate the average increase:

• At $x = 0$, $y = 65$.
• At $x = 4$, $y = 83$.

Compute the average rate of change (slope estimate):

So on average, the score increases about 4.5 points per hour studied.

Your intercept estimate says $b = 65$, and your slope estimate is $m \approx 4.5$. Among the answer choices, the equation whose slope and intercept match these values is $y = 4.5x + 65$.