A quadratic function graphs as a parabola, a symmetric U-shaped curve.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Which way it opens
A positive leading coefficient opens upward, giving a minimum. A negative one opens downward, giving a maximum.
The vertex
The vertex is the turning point. For y = ax² + bx + c it sits at x = −b ÷ 2a.
The axis of symmetry
A vertical line through the vertex mirrors the curve. Two points at equal distance from it share a height.
Intercepts
The y-intercept is c. The x-intercepts, where the curve crosses the axis, are the roots, and there may be two, one or none.
The size of a
A larger absolute value of a makes a narrower parabola. Values between −1 and 1 widen it.
The features worth naming
Vertex, axis of symmetry, intercepts and direction of opening. Those four describe a parabola completely, and every form of the equation makes one or two of them easy to read.
The leading coefficient sets the direction
Positive a opens upward and gives a minimum; negative a opens downward and gives a maximum. Knowing which before calculating tells you whether the vertex should be the highest or lowest point.
Symmetry halves the work
Every parabola is symmetric about a vertical line through its vertex. Plot the vertex and one point either side at equal distances and the y-values match, giving a good sketch from three points.
Roots are x-intercepts
Solving ax² + bx + c = 0 finds exactly where the graph meets the x-axis. Solving an equation and finding intercepts are the same task, which is why a sketch helps before any algebra.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Rosa Locate a Vertex
Rosa analyses y = x² − 4x + 3.
- Step 1
The leading coefficient is 1, positive, so the parabola opens upward.
Step 4: Your Turn
Practice makes it stick.
The Turning Point
Problem 1 of 2
y = x² − 4x + 3. What is the x-coordinate of the vertex?
The Intercept
Problem 2 of 2
y = x² − 4x + 3. What is the y-intercept?
Read the Parabola
1 of 8
y = x² − 6x + 5. Vertex x-coordinate?
2 of 8
y = 2x² + 8x. Vertex x-coordinate?
3 of 8
y = −x² + 4. Does it open up or down? 1 up, 2 down.
4 of 8
y = 3x² − 12x + 7. What is the y-intercept?
5 of 8
y = x² − 6x + 5 at x = 3. What is y?
6 of 8
A parabola opening upward has a maximum or minimum? 1 maximum, 2 minimum.
7 of 8
Sort each coefficient by what it controls.
Tap something to move it.
- Empty
- Empty
8 of 8
y = x² − 10x + 2. Vertex x-coordinate?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = x² − 8x + 1. What is the vertex x-coordinate?
Question 2 of 2
What does a negative leading coefficient mean?
What You Learned
- A quadratic graphs as a symmetric parabola.
- The vertex sits at x = −b ÷ 2a, and c is the y-intercept.
- The sign of a decides the direction; its size decides the width.