The process of plotting a quadratic function by identifying its key features: vertex, axis of symmetry, direction of opening, y-intercept, and x-intercepts (if they exist).
A parabola is a U-shaped curve (or upside-down U). Start by finding the vertex—that is the turning point. Then the axis of symmetry tells you the curve is a mirror image on both sides. Plot a few symmetric points and connect them in a smooth curve.
Showing a random 20 of 50 problems.
Example 1
easy
Find the axis of symmetry of y=x2+8x+1.
Example 2
hard
Find the vertex of y=3x2+12x+5 and state max or min value.
Example 3
easy
Find the x-intercepts of y=x2−9.
Example 4
challenge
A parabola has axis of symmetry x=1 and passes through (3,0) and (0,6). Find its equation in factored form.
Example 5
medium
By symmetry, if (1,7) lies on a parabola with axis x=4, what other point on the parabola has the same y-value?
Example 6
medium
Sketch direction: does y=0.1x2 open more or less steeply than y=x2?
Example 7
easy
Does f(x)=5x2+1 have any x-intercepts?
Example 8
easy
Does y=−4x2+1 open up or down?
Example 9
easy
Find the y-intercept of y=2x2−x+7.
Example 10
easy
The vertex of y=(x−1)2+2 is a minimum or maximum?
Example 11
hard
Sketch the key features of y=−2(x−1)2+8 and find its x-intercepts.Key features of y = −2(x−1)² + 8
Example 12
challenge
The graph of y=ax2 passes through (2,12). Find a and the y-value at x=−2.
Example 13
hard
Where do y=x2 and y=x+2 intersect?Intersections of y = x² and y
Example 14
easy
Find the vertex and y-intercept of y=−x2+4x.Vertex and y-intercept of y = −x² + 4x
Example 15
challenge
For what values of c does y=x2−6x+c have its vertex on the x-axis?
Example 16
medium
Sketching y=x2+2x+1: how many x-intercepts?
Example 17
easy
Find the vertex of y=x2−6x+5.Find the vertex of y
Example 18
medium
Find the vertex and direction of y=−2x2+8x−5.Vertex and direction of y