Mastering Linear Functions for the SAT

Linear functions are a fundamental concept in algebra and appear frequently on the SAT. Understanding linear functions is crucial for success on the test, as they form the foundation for more complex mathematical concepts. This lesson will guide you through the basics of linear functions, how to work with them, and how to solve SAT problems involving linear functions. By the end of this lesson, you'll be equipped with the knowledge and skills to tackle any linear function problem on the SAT with confidence.

A linear function is a mathematical relationship where the graph forms a straight line. The standard form of a linear function is

$f(x)=mx+b$ or $y=mx+b$

where:

• $m$ is the slope of the line, which measures the steepness and direction of the line
• $b$ is the y-intercept, which is the point where the line crosses the y-axis

The slope $m$ represents the rate of change - how much $y$ changes when $x$ increases by 1 unit. If $m$ is positive, the line rises from left to right. If $m$ is negative, the line falls from left to right. If $m=0$, the line is horizontal.

Linear functions can also be written in other forms:

• Point-slope form: $y-y_1=m(x-x_1)$, where $(x_1,y_1)$ is a point on the line
• Standard form: $Ax+By=C$, where A, B, and C are constants

On the SAT, you'll need to recognize linear functions, find slopes and y-intercepts, graph linear functions, and solve problems involving linear relationships.

Working with linear functions on the SAT involves several key skills:

1. Finding the slope:

   • From two points: $m=\frac{y_2-y_1}{x_2-x_1}$
   • From the equation: Rewrite in slope-intercept form $y=mx+b$ to identify $m$

2. Finding the y-intercept:

   • From the equation: Rewrite in slope-intercept form to identify $b$
   • From a point and slope: Use $b=y-mx$

3. Graphing linear functions:

   • Plot the y-intercept $(0,b)$
   • Use the slope to find additional points
   • Connect the points with a straight line

4. Writing equations of lines:

   • With slope and y-intercept: Use $y=mx+b$
   • With slope and a point: Use point-slope form $y-y_1=m(x-x_1)$
   • With two points: Find the slope first, then use point-slope form

5. Interpreting linear functions in context:

   • The slope represents the rate of change (e.g., cost per item, speed)
   • The y-intercept represents the initial value (e.g., fixed cost, starting point)

6. Solving systems of linear equations:

   • Substitution method
   • Elimination method
   • Graphing method

Remember to always check your work by substituting your answer back into the original equation.

Part 1: Identifying Slopes and Y-intercepts

For each equation, identify the slope and y-intercept:

1. $y=3x+4$
2. $y=-2x+7$
3. $2x+3y=12$
4. $4x-5y=10$
5. $y=\frac{1}{2}x-3$

Part 2: Finding Equations of Lines

Write the equation of each line in slope-intercept form:

1. Slope = 4, y-intercept = -2
2. Slope = -1/3, passing through the point (6, 2)
3. Passing through the points (1, 3) and (4, 9)
4. Slope = 0, passing through the point (5, 7)
5. Perpendicular to $y=2x+1$ and passing through the point (3, 4)

Part 3: Graphing Linear Functions

Graph each of the following linear functions:

1. $y=2x-3$
2. $y=-x+4$
3. $3x+2y=6$
4. $y=-\frac{1}{2}x+1$

Part 4: Word Problems

1. A taxi charges $2.50forthefirstmileand$1.75 for each additional mile. Write a linear function that gives the cost $C(m)$ for a ride of $m$ miles.

2. A car's value depreciates linearly from $25,000whennewto$5,000 after 10 years. Write a linear function that gives the car's value $V(t)$ after $t$ years.

1. Confusing the slope and y-intercept: Students often mix up which number is the slope and which is the y-intercept in the equation $y=mx+b$. Remember that $m$ (the coefficient of $x$) is the slope, and $b$ (the constant term) is the y-intercept.

2. Calculating slope incorrectly: When finding the slope between two points, students sometimes divide the change in x by the change in y instead of the change in y by the change in x. Always use $m=\frac{y_2-y_1}{x_2-x_1}$.

3. Misinterpreting negative slopes: A negative slope means the line falls from left to right, not that it's decreasing in value. The value of $y$ decreases as $x$ increases.

4. Confusing horizontal and vertical lines: A horizontal line has equation $y=c$ (slope = 0), while a vertical line has equation $x=c$ (undefined slope). Vertical lines are not functions.

5. Misunderstanding parallel and perpendicular lines:

   • Parallel lines have the same slope
   • Perpendicular lines have slopes that are negative reciprocals of each other (product of slopes = -1)

6. Forgetting to check solutions: Always verify your answers by substituting back into the original equations, especially when solving systems of equations.

7. Misinterpreting the meaning of slope in context: In real-world problems, the slope represents the rate of change (e.g., cost per item, speed, growth rate). Students sometimes fail to connect the mathematical concept to its real-world meaning.

8. Assuming all relationships are linear: Not all relationships between variables are linear. On the SAT, be careful to check whether a linear model is appropriate for the given context.