Quadratic functions: roots, vertex, and the shape of a parabola

Where we are on the map

We move from straight-line change to a curved but precisely described relationship.

Begin with a familiar image

The height of a thrown object first rises and then falls. A parabola can model this change where a line cannot.

Precise meaning

The graph of y=ax²+bx+c is a parabola. Roots occur at y=0, the vertex gives a minimum or maximum, and the sign of a sets the opening direction.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

standard form → roots from factors → vertex from completed square → graph check

Worked example

x²−4x+3=(x−1)(x−3)=(x−2)²−1; roots are 1 and 3, vertex is (2,−1).

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Calling the vertex (−2,−1) is wrong: (x−2)² reaches its minimum when x=2.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Factor and complete the square for y=x²−6x+5; find its roots and vertex.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

In an exponential function, a constant factor replaces a constant additive step.

Course contents