Quadratic Functions and Intersections with a Straight Line

Investigate how a, b, and c affect the quadratic graph y = ax² + bx + c, where a ≠ 0. Explore its intersections with the straight line y = mx + k.

Graph area

y = x²

Straight-line controls

Straight line: y = x

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Intersection points

Investigative Activities

    Investigation 1: Move the Straight Line

    Keep \(a=1\), \(b=0\), \(c=0\), and \(m=1\). Your graphs are \(y=x^2\) and \(y=x+k\).

    1. Start with k = 0. Where do the graphs intersect? Check your answer by solving x² = x.
    2. Move k to 2. Describe how the line moves and what happens to the distance between the intersection points.
    3. Slowly decrease k. Find the value where the two intersections merge into one. Record that value and the intersection coordinates.
    4. Decrease k further. Explain why there are now no real intersections.
    5. Complete this statement: There are two intersections when k is ______, one intersection when k is ______, and no intersections when k is ______.
    Reveal the mathematical explanation

    At an intersection, x² = x + k. Rearranging gives x² − x − k = 0.

    The discriminant is D = (−1)² − 4(1)(−k) = 1 + 4k.

    • Two intersections when k > −0.25.
    • One intersection when k = −0.25.
    • No real intersections when k < −0.25.

    At k = −0.25, the line is tangent to the parabola at (0.5, 0.25).

    Investigation 2: Keep the Line Fixed, Change the Quadratic Coefficients

    Learning goal

    Investigate how changing a, b, or c affects a parabola and its intersections with a fixed straight line. Use the graph to make conjectures, then use algebra to explain your findings.

    Before you begin

    1. Set m = 1 and k = 0. Leave these controls unchanged throughout the activity. Your straight line is y = x.
    2. Start with a = 1, b = 0, and c = 0. Your parabola is y = x².
    3. Change only one quadratic coefficient at a time. Reset the other coefficients as instructed.
    4. Predict before moving a slider. Record the settings, your observation, and a mathematical explanation. Coordinates displayed by the applet may be rounded.

    Part A: Change a — Opening, Width, and Intersections

    Keep b = 0 and c = 0. The equations are y = ax² and y = x.

    1. Compare a = 0.5, a = 1, and a = 2. Predict which parabola will be widest, then test your prediction.
    2. Try a = −1 and a = −2. What changes when a becomes negative? What happens to the vertex?
    3. For each value of a, record both intersection points. Which intersection stays fixed?
    4. Can changing only a produce no intersections with this line? Explain your reasoning.
    Part A: Reveal the explanation

    When b = 0 and c = 0, the vertex stays at (0, 0). Positive a makes the parabola open upward; negative a makes it open downward. Increasing the magnitude of a makes the parabola narrower.

    At an intersection, both equations have the same y-value:

    ax² = x

    Rearrange and factor:

    ax² − x = x(ax − 1) = 0

    Therefore x = 0 or x = 1/a. Because the line is y = x, each intersection has equal x- and y-coordinates:

    (0, 0) and (1/a, 1/a).

    For example, a = 2 gives (0, 0) and (0.5, 0.5). The value a = −1 gives (0, 0) and (−1, −1).

    Since a is nonzero, 1/a cannot equal 0. There are always two distinct intersections under these conditions, even if one is outside the visible graph window.

    Part B: Change b — Moving the Vertex

    Reset a = 1 and c = 0. Keep the line y = x unchanged. The parabola is now y = x² + bx.

    1. Try b = −2, 0, 1, 2, and 4. Describe how the vertex moves. Does it move only horizontally?
    2. Compare b = −2 and b = 2. How are the two parabolas related?
    3. Why does every parabola in this part pass through the origin?
    4. Record the intersection points. Predict the value of b at which they merge, then test your prediction.
    5. At the merging point, does the line cross the parabola or touch it? Justify your answer using algebra.
    Part B: Reveal the explanation

    The vertex of y = x² + bx is (−b/2, −b²/4). Changing b generally changes both vertex coordinates, so this is not simply a horizontal translation.

    Opposite values of b produce parabolas that are reflections of each other across the y-axis. All pass through (0, 0), because substituting x = 0 gives y = 0.

    To find intersections with y = x:

    x² + bx = x

    x² + (b − 1)x = 0

    x(x + b − 1) = 0

    Thus x = 0 or x = 1 − b. The intersection points are:

    (0, 0) and (1 − b, 1 − b).

    When b = 1, both solutions become x = 0. The intersection equation becomes x² = 0, giving one repeated root. The line is tangent to the parabola at the origin.

    Final Design Challenge

    Keep the straight line y = x fixed. Create two different parabolas that each touch this line at exactly one point. You may now change more than one quadratic coefficient.

    1. Record both quadratic equations.
    2. Record their tangent points.
    3. Prove that each has exactly one intersection by forming and solving its intersection equation.
    4. Describe what you changed and why your method worked.

    Your Conclusion

    Explain how a, b, and c affected the parabola in these experiments. State which conclusions depended on keeping the other coefficients fixed.

  1. Change one quadratic coefficient at a time and describe what happens to the parabola.
  2. Investigate when the straight line and parabola have two intersections, one intersection, or no intersections.