Linear Functions and Graphing
10 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

Which equation represents a linear function?

  • $y = 5x - 4$ (correct)
  • $y = rac{1}{x} + 7$
  • $y = ext{sin}(x)$
  • $y = 3x^2 + 2$

    • What is the slope of the line represented by the equation $y = 2x + 5$?

  • 2 (correct)
  • 5
  • 0
  • -2

    • Which of the following points lies on the line represented by the equation $y = 4x - 1$?

  • (1, 3)
  • (2, 5) (correct)
  • (3, 8)
  • (0, -1) (correct)

    • What is the y-intercept of the line given by the equation $y = -3x + 6$?

    <p>6 (C)</p> Signup and view all the answers

    What is the general form of a linear function?

    <p>$y = mx + b$ (B), $Ax + By + C = 0$ (C)</p> Signup and view all the answers

    If a linear function has a slope of $-2$ and passes through the point $(3, 4)$, what is the equation of the line?

    <p>$y = -2x + 10$ (C)</p> Signup and view all the answers

    What is the slope of the line represented by the equation $3x - 4y = 12$ when rewritten in slope-intercept form?

    <p>$-\frac{3}{4}$ (A)</p> Signup and view all the answers

    A linear function is represented by the equation $y = mx + 7$. If the value of $m$ is halved, what is the new y-intercept if the original slope is $4$?

    <p>$7$ (A)</p> Signup and view all the answers

    If the function $f(x) = 5x - 3$ is translated 2 units up, what is the new equation?

    <p>$f(x) = 5x + 2$ (A)</p> Signup and view all the answers

    Which of the following scenarios describes a linear function correctly?

    <p>The distance traveled over time with a constant speed. (B)</p> Signup and view all the answers

    Flashcards

    Linear Function

    A function whose graph is a straight line.

    Elementary Function

    A basic function used to describe relationships between variables.

    Freshman College

    First year of college, often concerning fundamental math concepts.

    Question

    A query or request for information.

    Signup and view all the flashcards

    Linear Function Question

    A question about a linear function, usually related to its properties.

    Signup and view all the flashcards

    Slope of a Linear Function

    The rate of change of a linear function, represented by 'm' in the equation y = mx + b. It determines how steep the line is and the direction it goes.

    Signup and view all the flashcards

    Y-intercept of a Linear Function

    The point where the line intersects the y-axis. It's represented by 'b' in the equation y = mx + b.

    Signup and view all the flashcards

    How to Find the y-intercept?

    Set x = 0 in the linear equation and solve for y. The resulting value of y is the y-intercept.

    Signup and view all the flashcards

    How to Find the Slope?

    Calculate the ratio of the difference in y-coordinates to the difference in x-coordinates between two points on the line. This is represented by 'm' in the equation y = mx + b.

    Signup and view all the flashcards

    Study Notes

    • A linear function is a function whose graph is a straight line. It can be represented by the equation y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

    Key Characteristics of Linear Functions

    • The slope (m) represents the rate of change of y with respect to x. A positive slope indicates an upward trend, a negative slope a downward trend, and a zero slope a horizontal line.

    • The y-intercept (b) is the point where the line crosses the y-axis. It represents the value of y when x is zero.

    • The equation is of the first degree (the highest power of the variable is 1).

    Graphing Linear Functions

    • To graph a linear function, two points are sufficient. Find two points that satisfy the equation (x, y) and plot them on a coordinate plane. Draw a straight line passing through these two points.

    • Alternatively, you can use the slope-intercept form (y = mx + b) directly: plot the y-intercept (0, b), then use the slope (m) to find a second point. Rise over run (m = rise/run) determines the change in y for a given change in x.

    Finding the Equation of a Line

    • Given two points (x₁, y₁) and (x₂, y₂), the slope can be calculated as m = (y₂ - y₁)/(x₂ - x₁)

    • After finding the slope, use the point-slope form, y - y₁ = m(x - x₁), to find the equation. Substitute one of the points and the slope into this form.

    Special Cases of Linear Functions

    • Horizontal lines: These have a slope of zero, and their equation is of the form y = b.

    • Vertical lines: These have an undefined slope (division by zero) and their equation is of the form x = a.

    Applications of Linear Functions

    • Many real-world situations involve linear relationships. Examples include:
      • Calculating total cost (if there's a constant rate like cost per item).
      • Determining distance traveled if speed is constant.
      • Modeling growth or decay at a fixed rate.

    Determining the Slope and Y-Intercept

    • Given an equation in the form y = mx + b: the slope is 'm', and the y-intercept is 'b'.

    • Sometimes an equation is given in a different form and needs to be rearranged to the slope-intercept form. For example, if you are given 2x + 3y = 6, rearrange to isolate 'y'.

    Example Problems (Illustrative)

    • Find the slope and y-intercept of the line described by the equation 2x - 4y = 8.

      • Rearrange the equation to slope-intercept form (y = mx + b):

        • 2x - 8 = 4y
        • y = (1/2)x -2

      • The slope is 1/2 and the y-intercept is -2.

    • Find the equation of the line passing through the points (3, 1) and (-1, 5).

      • Calculate the slope: m = (5 - 1) / (-1 - 3) = -1.
      • Use the point-slope form: y - 1 = -1(x - 3)
        • Simplify to get the equation: y = -x + 4

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Explore the key characteristics of linear functions, including slope and y-intercept. Learn how to graph these functions using two points and the slope-intercept form. Test your understanding of the concepts applicable to linear equations.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser