Evaluating a Linear Function

Looking for free SAT practice questions that test your understanding of linear functions? This step-by-step solution explores how to deduce a linear function with information about two points. You will learn how to use the two points to find the slope and subsequently the linear function.

Question

For a linear function 𝑓, it is given that 𝑓(2) = 7 and 𝑓(6) = 19. What is the value of 𝑓(10)?

1. 22
2. 28
3. 31
4. 34

To solve this free SAT Algebra practice question, let's list down what is known from the question

1. What are we given?
   • f is a linear function, which means its graph is a straight line and follows the form: f(x) = mx + b
   • f(2) = 7
   • f(6) = 19
2. What can we do with the two values of the function?
   • f(2) = 7 for a linear function means when x = 2, f(x) or y = 7. So coordinates of a point the line passes through is (2, 7)
   • And the coordinates of the second point that the line passes through is (6, 19)
3. What can we do with the coordinate pairs?:
   • You guessed it — we can find the slope of the line using these two points.
   • If the coordinates of the two points are (x₁, y₁) and (x₂, y₂), the slope = \\frac{y_2 - y_1}{x_2 - x_1})
   • Therefore, slope 'm' of this line = \\frac{19 - 7}{6 - 2}) = \\frac{12}{4}) = 3
4. What after finding the slope? Let's set about to find the y-intercept b:
   • Now we use that slope to find the y-intercept b using one of the points — let's use (2, 7):
   • f(x) = 3x + b ⇒ 7 = 3(2) + b
   • ⇒ 7 = 6 + b ⇒ b = 1
   • So, the function is 3x + 1
5. What is 𝑓(10):
   • 𝑓(10) = 3(10) + 1 = 31

Option C is the Correct Answer