A. Characteristics of Functions · Quadratic Functions

Lines That Touch, Cross or Miss: Linear-Quadratic Problems

A2.5 A2.1About 70 minutes

Learning goals

I can find the values of a parameter for which a line meets a parabola twice, once or never.
I can find a tangent line with a given slope or through a given point, and its point of tangency.
I can solve real-world problems about whether a straight path meets a curved one.
I can find the minimum vertical gap between a line and a parabola.

Words to know

tangent line
A line that meets a parabola at exactly one point (for lines that are not vertical); the discriminant of the combined equation is 0.

family of lines
Lines that share a property, such as y = 2x + b (all with slope 2) or y = mx − 1 (all through (0, −1)).

vertical gap
The difference in y-values of two graphs at the same x.

Touching, crossing or missing

In the last lesson you found where a given line meets a given parabola. Now the line can change. Imagine a family of parallel lines, all with slope 2, sliding up and down across a parabola. Low lines cross it twice, one special line just touches it, and higher lines miss it completely.

This is the curriculum's own investigation: determine the equations of the lines that have a slope of 2 and that intersect the quadratic function f(x) = x(6 − x) once, twice, never. The discriminant answers it in a few lines of algebra.

Worked example

Example 1: The curriculum investigation

Lines with slope 2 have equations y = 2x + b. For which values of b does the line meet f(x) = x(6 − x) once, twice or never?

Set equal: 6x − x² = 2x + b.
Rearrange: −x² + 4x − b = 0, or x² − 4x + b = 0.
D = (−4)² − 4(1)(b) = 16 − 4b.
Once: 16 − 4b = 0, so b = 4. The line y = 2x + 4 is tangent. Then x² − 4x + 4 = (x − 2)² = 0 gives x = 2, and the point is (2, 8).
Twice: 16 − 4b > 0, so b < 4.
Never: 16 − 4b < 0, so b > 4.

y = 2x + b meets f twice when b < 4, once when b = 4 (tangent at (2, 8)), and never when b > 4.

Three lines with slope 2 and the parabola f(x) = x(6 − x). y = 2x (b = 0) crosses twice, at (0, 0) and (4, 8). y = 2x + 4 touches at (2, 8). y = 2x + 7 misses.

The recipe

Write the line with an unknown (y = mx + b with m or b unknown).
Set it equal to the quadratic and rearrange to ax² + bx + c = 0.
Write the discriminant in terms of the unknown.
Solve D = 0 (tangent), D > 0 (two points) or D < 0 (no points).

Worked example

Example 2: Unknown slope, fixed point

Find the lines through (0, −1) that are tangent to y = x², and the points where they touch.

A line through (0, −1) is y = mx − 1.
x² = mx − 1, so x² − mx + 1 = 0.
Tangent: D = m² − 4 = 0, so m = 2 or m = −2.
m = 2: x² − 2x + 1 = (x − 1)² = 0, so x = 1 and the point is (1, 1).
m = −2: x² + 2x + 1 = (x + 1)² = 0, so x = −1 and the point is (−1, 1).

y = 2x − 1, touching at (1, 1), and y = −2x − 1, touching at (−1, 1).

Worked example

Example 3: Will the drone clear the hill?

A survey drone flies in a straight line, y = 0.5x + 30, over a hill whose cross-section is y = −0.01x² + 1.5x (all in metres). Does the drone's path ever meet the hill? If not, what is the closest vertical gap?

Set equal: −0.01x² + 1.5x = 0.5x + 30, so −0.01x² + x − 30 = 0.
Multiply by −100: x² − 100x + 3000 = 0.
D = 10 000 − 12 000 = −2000 < 0, so the path never meets the hill.
Vertical gap: d(x) = (0.5x + 30) − (−0.01x² + 1.5x) = 0.01x² − x + 30.
This is a quadratic with a minimum at x = −(−1)/(2 × 0.01) = 50. d(50) = 25 − 50 + 30 = 5.

No, the path never meets the hill. The closest vertical gap is 5 m, at x = 50 m.

The drone's path (dashed) stays above the hill (solid). The vertical gap is smallest, 5 m, at x = 50 m.

Worked example

Example 4: Lines through the origin

For what values of k does the line y = kx meet y = x² + 4 exactly once?

x² + 4 = kx, so x² − kx + 4 = 0.
D = k² − 16 = 0, so k = 4 or k = −4.
Points: k = 4 gives (x − 2)² = 0, so (2, 8); k = −4 gives (x + 2)² = 0, so (−2, 8).

k = ±4: y = 4x touches at (2, 8) and y = −4x touches at (−2, 8).

Hands-on activity

Tangent hunt

You need: graph paper, a pencil, a clear ruler and a pin or a sharp pencil point to pivot on (a grown-up can help with the pin).

Draw axes and plot y = x² carefully for x = −3 to 3 (1 square = 1 unit).
Mark the point (0, −1) on the y-axis.
Hold the ruler's edge against the pencil point at (0, −1) and slowly rotate it. Count how many times the edge crosses the parabola as you rotate.
Find the two positions where the edge just touches the parabola. Read the touching points and estimate the slopes.
Compare with Example 2.

What to look for: steep lines cross twice, shallow lines (close to horizontal) miss the parabola, and the two boundary lines touch at about (1, 1) and (−1, 1), with slopes 2 and −2.

Just for fun

Tangent lines are the world's most polite guests: they visit the parabola exactly once and never cross it.

Your turn

For what value of b is y = x + b tangent to y = x²?

For what values of b does y = −2x + b meet f(x) = x² + 2x twice?

Find the line with slope 4 that is tangent to y = x² + 1, and the point of tangency.

For what values of k does y = kx meet y = x² + 9 exactly once?

Does y = 3x − 2 meet f(x) = −x² + x + 1? If so, where?

Using Example 1, how many times does y = 2x + 1 meet f(x) = x(6 − x)?

More practice: warm-up

What does D = 0 mean for a line and a parabola?

Set y = x + b equal to y = x² and write the quadratic equation in standard form.

How many times does y = 5 meet f(x) = −(x − 1)² + 5?

How many times does y = 6 meet f(x) = −(x − 1)² + 5?

How many times does y = 2 meet f(x) = −(x − 1)² + 5?

More practice: on your own

Find the values of m for which y = mx + 1 is tangent to y = x² + 2.

Find b so that y = −2x + b is tangent to f(x) = −x² + 4x. Find the point of tangency.

A straight road is planned along y = 0.1x + 8 through a hill whose cross-section is y = −0.02x² + 1.3x (metres). Does the road meet the hill? Where?

A zipline cable follows y = −0.5x + 20 above a snowy mound y = −0.1x² + 2x (metres). Does the cable touch the mound? What is the smallest vertical gap?

Explain why, for a parabola that opens up or down, exactly one line with a given slope is tangent to it.

Challenge

For which values of b does y = 2x + b intersect y = x² − 4x + 7 twice? Find the tangent line and its point of tangency.

Find the two lines through (0, −4) that are tangent to y = x², and their points of tangency.

Find c so that the line y = x + 1 is tangent to y = x² + c, and the point of tangency.

Check your understanding

Quiz

Lines y = 2x + b meet f(x) = x(6 − x) exactly once when b =

Lines y = 2x + b meet f(x) = x(6 − x) twice when:

The lines through (0, −1) that are tangent to y = x² have slopes:

A road and a hill give an equation with a negative discriminant. This means:

For what b is y = x + b tangent to y = x²?

Which line through the origin is tangent to y = x² + 4?

For parents and teachers

Big idea: Expectation A2.5, using the discriminant (A2.1) to decide when a line meets a quadratic function once, twice or never. Example 1 is the curriculum's own sample investigation.

How to help: Graphing technology (a free online graphing calculator) is great here: add a slider for b in y = 2x + b and watch the line move across f(x) = x(6 − x). Ask your teen to predict the tangent value before moving the slider.

Common difficulties:

Rearranging the combined equation with sign errors, which changes the discriminant.
Mixing up the letter b in y = mx + b with the b in ax² + bx + c. Some students rename the line's intercept to k to avoid this.
Forgetting to find the point of tangency after finding the line.
In Grade 12 Advanced Functions (MHF4U) and Calculus and Vectors (MCV4U), tangent lines are found with rates of change and derivatives; this lesson is an early, algebraic look at the same idea.