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2.3 – Linear Functions

Objectives

  • Describe linear functions with words, graphically, with a table of values, or with an algebraic expression.
  • Identify and interpret the slope and intercept of a line.
  • Write equations of lines.

 

Key Terms & Notes

Axes & Intercepts
  • x-axis: The horizontal axis in a Cartesian coordinate system.
  • y-axis: The vertical axis in a Cartesian coordinate system.
  • x-intercept: A point where the graph of a function crosses the x-axis.
    • This is the graph coordinate (x, 0).
  • y-intercept: A point where the graph of a function crosses the y-axis.
    • This is the graph coordinate (0, y).
    • A function has at most one y-intercept (remember the vertical line test).
    • The y-intercept of the line with equation y = mx + b is the point (0, b).
Linear Functions
  • A linear function is a function whose graph is a line.
  • A line is written in three forms:
    • Standard form: ax + by = c
    • Slope-Intercept form: y = mx + b
    • Point-Slope form: y-y_1=m(x-x_1)

 

Slope
  • Slope is the measure of the steepness of a line.
  • Slope equals rise divided by run for any two points on the line.
    • Rise over run, written as \frac{rise}{run}
    • Change (\Delta) in y over change (\Delta) x, written as \frac{\Delta y}{\Delta x}

Alg 2A - Slope


  • Using slope to write point-slope
    • Once you find the slope, put it into the point-slope form equation.
    • You can use ANY point on the line to write the formula for this line in point-slope form.

Alg 2A - point-slope


  • Rise and Fall
    • A line that rises from left to right has a positive slope.
    • A line that falls from left to right has a negative slope.
    • A line that is horizontal has a zero slope.
    • A line that is vertical has an undefined slope.
      • You can’t divide a number (the rise) by zero (the run).
Parallel & Perpendicular Lines
  • Parallel lines have equal slopes.
    • Ex. The following lines are parallel because they have the same slope.
      • y=3x+4
      • y=3x-2
  • Perpendicular lines have negative reciprocal slopes.
    • Ex. The following lines are perpendicular because their slopes are negative reciprocals of one another.  This means that the slopes are flipped and one is positive while the other is negative.
      • y=3x+4
      • y=- \frac{1}{3}-2

 

Examples

  • Ex 1. What is the slope of a line is parallel to y=\frac{1}{2}x+3?
    • Answer: \frac{1}{2}
  • Ex 2. What is the graph of the line y=-3x-2?
    • Answer: There is a negative slope and a negative y-intercept, so the line crosses the y-axis at -2 and slopes down from left to right.

Alg2A 02.03 Ex02

  • Ex 3. What is the slope of the function, represented by the table of values below?

Alg2A 02.03 Ex03

Answer: Pick any two points in the chart and use the slope formula: \frac{y_2-y_1}{x_2-x_1}

\frac{6-0}{0-3}=\frac{6}{-3}=-2

  • Ex 4. Which of the following is the equation of a line that passes through the point (1,4) and is parallel to the x-axis?
    • Answer: The x-axis has a zero slope, so substitute into the point-slope form: y-4=0(x-1).
    • This becomes y-4=0, which simplifies to y=4.
  • Ex 5. Which of the following is the equation of a line that passes through the points (0,6) and (2,10)?
    • To solve this, do the following steps:
      1. Find the slope: m=\frac{10-6}{2-0}, which simplifies to \frac{4}{2}=2.
      2. Put the slope and ONE of the points into point-slope form: y-6=2(x-0).
      3. Use distribution and inverse operations to convert to slope-intercept form: y-6=2x, and then again to y=2x+6.
    • Answer: y=2x+6
  • Ex 6. Jenny is selling books during the summer to earn money for college. She earns a commission on each sale but has to pay for her own expenses. After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on. She writes the equation of the function like this: E(x)=20x-50, where x is the number of doors she knocks on during the week and E(x) is her earnings for the week in dollars.
  • What does the slope of Jenny’s function represent?
    • Answer: for each additional door she knocks on, her earnings will increase by $20.
    • This is because the slope represents how much money she makes PER (each) door she knocks on.  For every door, she makes $20 more dollars.  You can multiply 20 times the number of doors knocked on.  That’s why it is written as 20 times x.
    • The $50 represents her base salary (and therefore, her expenses) for the week… it is not the slope.  If she doesn’t knock on any doors, she will actually lose (minus) the $50 and not be able to pay for her expenses.
  • Ex 7. The ordered pairs below represent a relation between x and y.
    • (-3,0), (-2,4), (-1,8), (0,12), (1,16), (2,20)
    • Could this set of ordered pairs have been generated by a linear function?
      • Answer: Yes, because the relative difference between y-values and x-values is the same no matter which pairs of (x, y) values you use to calculate it.
      • In other words, the slope is the same for any two points you calculate.  So, you have a line, which is represented by a linear function!

 

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