Question

the graph of a line is shown below.
which three of the following statements are true about the graph?
a. the constant of proportionality is \\(\frac{1}{2}\\).
b. the point (10, -10) would be included on the line.
c. the line has a negative k - value.
d. the line represents a direct variation relationship.
e. the line is horizontal.

Explanation:

Step1: Find the slope (constant of proportionality)

The line passes through \((0,0)\) and \((4,2)\). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 0}{4 - 0}=\frac{2}{4}=\frac{1}{2}\). So the constant of proportionality is \(\frac{1}{2}\), so the first statement is true.

Step2: Check if \((10, - 10)\) is on the line

Using the equation \(y=\frac{1}{2}x\) (since it's a proportional line through the origin), when \(x = 10\), \(y=\frac{1}{2}(10)=5
eq - 10\), so \((10,-10)\) is not on the line, the second statement is false.

Step3: Determine the value of \(k\) (slope)

For a proportional line \(y = kx\), \(k\) is the slope. We found the slope is \(\frac{1}{2}\), so \(k=\frac{1}{2}\), the third statement is true (assuming "mayme" is a typo for "may be" and \(k\) refers to the slope).

Step4: Check the type of variation

Since the line passes through the origin and has the form \(y = kx\), it represents a direct variation relationship, the fourth statement is true.

Step5: Check if the line is horizontal

A horizontal line has a slope of \(0\), but our slope is \(\frac{1}{2}
eq0\), so the line is not horizontal, the fifth statement is false.

Answer:

The true statements are: "The constant of proportionality is \(\frac{1}{2}\)", "The line may be a \(k\)-value (with \(k = \frac{1}{2}\))", "The line represents a direct variation relationship" (assuming the intended statements are as interpreted from the context). If we consider the options (assuming the first option is "The constant of proportionality is \(\frac{1}{2}\)", third is "The line may be a \(k\)-value" (with correct \(k\)), fourth is "The line represents a direct variation relationship"), these three are true.