Question

at a glance, kendra believes that the function represented on the graph is linear.
how can kendra determine if the function is actually linear?
○ she can check to see if the rate of vertical increase equals the rate of horizontal increase between each pair of points.
○ she can check to see if the sum of each y - value and x - value in every ordered pair is the same.
○ she can check to see if the quotient of each y - value and x - value in every ordered pair is the same.
○ she can check to see if the rate of change between the first two ordered pairs is the same as the rate of change between the first and last ordered pairs.

Explanation:

To determine if a function is linear, we check the rate of change (slope) between points. The rate of change formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For a linear function, the rate of change between any two pairs of points should be the same. Let's analyze each option:

• Option 1: Rate of vertical increase (change in y) and horizontal increase (change in x) being equal would mean slope = 1, but linear functions can have any constant slope, not just 1. So this is incorrect.
• Option 2: Checking if $x + y$ is the same is not a valid test for linearity. For example, the points here: $2 + 4 = 6$, $3 + 9 = 12$, $4 + 16 = 20$ – not equal, but this doesn't mean it's not linear (though in this case, the graph is curved, but the reasoning about the option is wrong).
• Option 3: Checking if $\frac{y}{x}$ is the same is for proportional relationships (linear through the origin), not all linear functions. Here, $\frac{4}{2}=2$, $\frac{9}{3}=3$, $\frac{16}{4}=4$ – not equal, but a linear function doesn't need $\frac{y}{x}$ constant unless it's proportional.
• Option 4: For a linear function, the rate of change between the first two points and between the first and last points (or any two pairs) should be equal. Let's calculate the rate of change between (2,4) and (3,9): $m_1=\frac{9 - 4}{3 - 2}=5$. Between (2,4) and (4,16): $m_2=\frac{16 - 4}{4 - 2}=\frac{12}{2}=6$. Wait, but the graph is actually curved (exponential maybe, since the points look like $y = x^2$: $2^2=4$, $3^2=9$, $4^2=16$). But the option's logic is about the test for linearity: if the rate of change between different pairs is the same, it's linear. So this option is the correct method to test linearity (even though in this case, the rates are different, showing it's not linear).

Answer:

D. She can check to see if the rate of change between the first two ordered pairs is the same as the rate of change between the first and last ordered pairs.