LEARN · EXPLAIN · REVISE

Find the minimum value over all real x. Enter the minimum value, not the x-coordinate.

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

TOPIC 01

Quadratic equations and functions

The required output is the vertical coordinate of the vertex. Entering −2 gives where the minimum occurs, not the minimum value.

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#Your turn

Find the minimum value over all real x. Enter the minimum value, not the x-coordinate.

f(x)=3x^2+12x+5

Working and explanation

BUILD THE REASONING

Hint 1

Factor 3 out of the terms containing x.

Hint 2

Use x² + 4x = (x + 2)² − 4.

Worked solution

Complete the square.
$f(x)=3[(x+2)^2-4]+5=3(x+2)^2-7$
Set the nonnegative square to zero.
$f(-2)=-7=\min f$

The minimum is −7, reached at x = −2.

Checks and common pitfalls: The required output is the vertical coordinate of the vertex. Entering −2 gives where the minimum occurs, not the minimum value.

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