Which equation matches the graph?

"m" represents the slope, which is the measure of steepness and orientation of a nonvertical line.
"b" represents the y-coordinate of the y-intercept, (0, b). The y-intercept is the point on the graph where the line intersects the y-axis. It is also the point where the x-coordinate equals to zero.

How to solve for the slope?

We can solve for the slope by using the following formula:

[tex]\displaystyle\mathsf{Slope\:(m)\:=\:\frac{Rise}{Run}\:=\Bigg(\frac{y_2-y_1}{x_2 - x_1}\Bigg)}[/tex]

Choose two points from the graph and substitute their values into the slope formula.

(x₁, y₁) = (2, 2)
(x₂, y₂) = (0,-4)

[tex]\displaystyle\mathsf{Slope\:(m)\:=\Bigg(\frac{y_2-y_1}{x_2 - x_1}\Bigg)\:=\:\Bigg(\frac{2-(-4)}{2-0}\Bigg)\:=\:\Bigg(\frac{2+4}{2}\Bigg)\:=\:\frac{6}{2}\:\:=\:3}[/tex]

Therefore, the slope of the line is 3.

How to find the y-intercept (b)?

As defined in the previous section of this post, the y-intercept (0, b) is the point on the graph where it crosses the y-axis. We can see that the line intersects the y-axis at point (0, -4). This means that the value of our y-intercept, b, is -4.

Finding the y-intercept algebraically.

Alternatively, we can solve for the y-intercept (b) by choosing a point on the graph and substitute its values into the slope-intercept form.

We will use and substitute the values of (x₁, y₁) and the slope into the function to solve for b.

(x₁, y₁) = (2, 2)

Slope (m ) = 3

f(x) = mx + b

⇒ 2 = 3(2) + b

⇒ 2 = 6 + b

⇒ 2 - 6 = 6 - 6 + b

⇒ -4 = 0 + b

⇒ - 4 = b

Hence, the value of the y-intercept is: b = 4.

Final Answer:

Based from our calculations, we can infer that the linear function that matches the graph is Option C: f(x) = 3x - 4.

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Keywords:

Linear functions

Linear equations

Slope-intercept form

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