LEARN · EXPLAIN · REVISE

Find the value of k for which the graph touches the x-axis at exactly one point. Give that point as well.

Read the idea, work independently, then explain what changed.

TOPIC 01

Quadratic equations and functions

For k < 9 there are two intersections; for k > 9 there are none. Changing k moves the graph vertically while its axis stays at x = 3.

PREDICT → MOVE → EXPLAIN

What does each coefficient change?

Before moving a slider, predict the opening, vertex and intercepts. Then compare the graph with your prediction.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Worked example

Find the value of k for which the graph touches the x-axis at exactly one point. Give that point as well.

y=x^2-6x+k

Working and explanation

BUILD THE REASONING

Hint 1

Touching the x-axis corresponds to a repeated real root.

Hint 2

Set the discriminant equal to zero, or make the minimum value zero.

Worked solution

Identify the coefficients and compute the discriminant.
$\Delta=(-6)^2-4(1)k=36-4k$
A repeated root requires a zero discriminant.
$36-4k=0\Rightarrow k=9$
Substitute the parameter and factor the expression.
$y=x^2-6x+9=(x-3)^2$

k = 9; the graph touches the x-axis at (3, 0).

Checks and common pitfalls: For k < 9 there are two intersections; for k > 9 there are none. Changing k moves the graph vertically while its axis stays at x = 3.

Think first. Reveal a hint when the class is ready.