Question

three points of a function are graphed.
which statement describes the function through the points?
the function is a direct variation function with a constant of variation of 1.5.
the function is a direct variation function with a constant of variation of 1.8.
the function is linear but is not a direct variation function.
the function is not a linear function.
(18, 30)
(14, 24)
(10, 18)

Explanation:

Step1: Check if the function is linear (calculate slopes)

First, calculate the slope between \((10, 18)\) and \((14, 24)\). The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
So, \(m_1=\frac{24 - 18}{14 - 10}=\frac{6}{4} = 1.5\).

Step2: Calculate slope between \((14, 24)\) and \((18, 30)\)

Using the slope formula again, \(m_2=\frac{30 - 24}{18 - 14}=\frac{6}{4}=1.5\). Since the slopes are equal, the function is linear.

Step3: Check if it's direct variation (passes through origin)

A direct variation function has the form \(y = kx\), which passes through \((0,0)\). Let's see if the line passes through \((0,0)\). Using the slope \(k = 1.5\), the equation is \(y=1.5x\). When \(x = 0\), \(y = 0\), but let's check with our points. For \((10,18)\), \(1.5\times10 = 15
eq18\). Wait, no—wait, we made a mistake. Wait, recalculate the slope again. Wait, \((10,18)\): \(y = 1.8x\)? Wait no, \(\frac{18}{10}=1.8\), \(\frac{24}{14}\approx1.714\), \(\frac{30}{18}\approx1.666\). Wait, no, earlier calculation was wrong. Wait, no: \((10,18)\): \(x = 10\), \(y = 18\); \((14,24)\): \(x = 14\), \(y = 24\); \((18,30)\): \(x = 18\), \(y = 30\). Let's recalculate the slope between \((10,18)\) and \((14,24)\): \(\frac{24 - 18}{14 - 10}=\frac{6}{4}=1.5\). Between \((14,24)\) and \((18,30)\): \(\frac{30 - 24}{18 - 14}=\frac{6}{4}=1.5\). So the slope is 1.5. Now, the equation of the line: using point-slope form. Let's take \((10,18)\), \(y - 18 = 1.5(x - 10)\). \(y=1.5x - 15 + 18=1.5x + 3\). So the y-intercept is 3, not 0. Therefore, it's a linear function but not direct variation (since direct variation has y-intercept 0). Wait, but wait, when we check the ratios: \(\frac{18}{10}=1.8\), \(\frac{24}{14}\approx1.714\), \(\frac{30}{18}\approx1.666\). Wait, no, the slope is 1.5, but the ratios of y to x are not constant. Wait, I'm confused. Wait, no—slope is 1.5, so the line is \(y = 1.5x + b\). Plugging in \((10,18)\): \(18 = 1.5\times10 + b\) → \(18 = 15 + b\) → \(b = 3\). So the line is \(y = 1.5x + 3\), which does not pass through the origin (since when \(x = 0\), \(y = 3\)), so it's linear but not direct variation. Wait, but earlier when I thought the slope was 1.5, but the ratios of y/x are not 1.5. Wait, \(\frac{18}{10}=1.8\), \(\frac{24}{14}\approx1.714\), \(\frac{30}{18}\approx1.666\). But the slope between the points is 1.5. So the function is linear (constant slope) but not direct variation (since it doesn't pass through the origin, and \(y/x\) is not constant).

Answer:

The function is linear but is not a direct variation function.