Representational Relationships of Lines and Parabolas

  • 7.5 Representational Relationships of Lines and Parabolas

    Grade: High School

    This two-part e-example allows for students to explore the effect of coefficients and constants within linear and quadratic functions. Options include the ability to the change the forms of equations and trace the graphs.




    On the upper right-hand corner, you can switch between exploring linear and quadratic functions.


    In the linear mode, you can use the double arrows to switch between the standard form and the point-slope form.

    Changing Values

    You can change the values of each coefficient by using the sliders on the left. You can link two or more sliders by clicking on the chains below each value. This will allow for both (or all three) values to increase or decrease by the same amount.


    You can trace how the coefficients change the graph and/or its properties by clicking on the line, curve, or vertexbutton (located in the lower left-hand corner). These buttons toggle such that once it is clicked on again, any traces are erased.

    Exploring the Graph

    You can explore coordinates of the graph by hovering over a specific point on the line or curve.


    1. How do changes in the values of m and b in the linear function f(x) = mx + b affect the graph? In the applet below, use the sliders to adjust m and b in f(x) = mx + b. How is the graph affected by changes in m? By changes in b?
    2. What are the effects on the graph of f(x) = mx + b if the values of m and b are changed simultaneously? Click on the Connect Sliders button. Adjust the sliders again. What happens to the equation and to the graph? Summarize your findings.
    3. What are the effects of changing m and b simultaneously for different lines? Uncheck the Connect Sliders button, and specify a different initial line. Then click on Connect Sliders again. Describe what happens. Compare your results to the results you obtained in task 2.
    4. Click on the Show Trace button. Predict starting values for m and b that, with the sliders connected, would result in the lines intersecting in—

    • Quadrant I,
    • Quadrant II,
    • Quadrant III,
    • Quadrant IV.

    Try out your values. Describe what happens.
    5. If m and b are simultaneously varied by the same amount, a family of lines is generated.

    1. Express the equation f(x) = mx + b so that it represents the family of lines. (Hint: Consider using k to represent the amount m and b are varied.)
    2. Use your equation to explain why all the lines meet at a common point and why the x-value at that common point is always –1.


    Objectives and Standards

    NCTM Standards and Expectations
    • Algebra
    • High School (9-12)
    • Algebra