Ximpledu

Quadratic Function

See how to solve a quadratic function.
17 examples and their solutions.

Quadratic Function: Opened Upward, Downward

Formula

y = ax2 + bx + c


a: (+)a: (-)
a: (+) → opened upward
a: (-) → opened downward
The shape of a quadratic function,
the U shape,
is called a parabola.

Example

y = x2 + 2x + 5
Upward / downward?
Solution

Example

y = -3x2 + x - 8
Upward / downward?
Solution

Quadratic Function: Axis of Symmetry

Formula

y = ax2 + bx + c


x = -b2a
The axis of symmetry is the line
that cuts the graph
into two symmetric pieces.

Example

y = x2 + 4x - 9
Axis of symmetry?
Solution

Example

y = -5x2 + x
Axis of symmetry?
Solution

Quadratic Function: Vertex

Formula

y = a(x - h)2 + k

The vertex (red point)
is the lowest point of the quadratic function.
(highest point: when a is (-).)
To find the vertex of a quadratic function,
change the quadratic function
to vertex form.
y = a(x - h)2 + k
Then the vertex is (h, k).
The vertex is on the axis of symmetry.
So x = h is the axis of symmetry.

Example

y = x2 - 4x + 5
Vertex?
Solution

Example

y = x2 + 6x - 1
Vertex?
Solution

Example

y = -x2 + 8x - 16
Vertex?
Solution

Quadratic Function: Finding Zeros

Formula

y = a(x - r1)(x - r2)

The zeros is the intersection
of the graph and the x-axis.
So, to find the zeros of the quadratic function,
1. Factor the quadratic function.
2. Set (left side) = 0.
3. Solve the quadratic equation: (left side) = 0.

Example

y = x2 - 2x - 3
Zeros?
Solution

Example

y = -3x2 + 12
Zeros?
Solution

Quadratic Function: Number of Zeros

Formula

y = ax2 + bx + c
→ D = b2 - 4ac


D > 0D = 0D < 0
Just like the D of a quadratic equation,
you can find the number of zeros
by using the D of the quadratic function.
(without solving the quadratic function)

Example

y = x2 + 8x - 3
Solution

Example

y = -4x2 + 4x - 1
Solution

Example

y = 2x2 - x + 7
Solution

Quadratic Inequality

Example

x2 - 3x - 10 ≤ 0
Solution

Example

x2 - 16 > 0
Solution

Example

-x2 + 10x - 25 ≥ 0
Solution

System of Equations: Quadratic-Linear

Example

y = x2 - 2x
y = x + 4
Solution

Number of Intersecting Points


D > 0D = 0D < 0
From a quadratic-linear system,
you'll get a quadratic equation.
The D of the quadratic equation determines
the number of the intersecting points.
D > 0: 2 intersecting points
D = 0: 1 intersecting point
D < 0: no intersecting points

Example

Find the range of k that makes the given functions intersect.
y = x2 - x
y = x + k
Solution