Graphing quadratic functions

Graphing Quadratic Functions

A quadratic function is a polynomial function of degree two. It can be expressed in the standard form:

f(x) = ax² + bx + c

where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola. This lesson will cover how to graph quadratic functions by identifying key features such as the vertex, axis of symmetry, and intercepts.

Understanding the Parabola

The shape of the parabola depends on the coefficient 'a'. If a > 0, the parabola opens upwards. If a < 0, it opens downwards. The vertex of the parabola is the highest or lowest point, depending on the direction it opens.

Finding the Vertex

To find the vertex of the quadratic function, you can use the formula for h:

h = -b/(2a)

Next, substitute h back into the function to find k:

k = f(h)

Example 1: Finding the Vertex

Consider the quadratic function:

f(x) = 2x² - 4x + 1

Here, a = 2 and b = -4. Let's find the vertex.

1. Calculate h:

h = -(-4)/(2 * 2) = 4/4 = 1

2. Substitute h into the function to find k:

k = f(1) = 2(1)² - 4(1) + 1 = 2 - 4 + 1 = -1

The vertex is (1, -1).

Finding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex. It has the equation:

x = h

In our example, since h = 1, the axis of symmetry is:

x = 1

Finding the x-intercepts

The x-intercepts are the points where the graph intersects the x-axis. To find the x-intercepts, set f(x) = 0 and solve for x:

0 = ax² + bx + c

Example 2: Finding the x-intercepts

Using our function:

0 = 2x² - 4x + 1

We can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Substituting a = 2, b = -4, and c = 1:

x = (4 ± √((-4)² - 4 * 2 * 1)) / (2 * 2)

Calculate the discriminant:

√(16 - 8) = √8 = 2√2

Now substitute back into the formula:

x = (4 ± 2√2) / 4 = 1 ± √2/2

The x-intercepts are:

x = 1 + √2/2 and x = 1 - √2/2.

Finding the y-intercept

The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, set x = 0 in the function:

f(0) = c

For our function, f(0) = 1. Thus, the y-intercept is (0, 1).

Sketching the Graph

Now that we have the vertex, axis of symmetry, x-intercepts, and y-intercept, we can sketch the graph.

1. Plot the vertex (1, -1).
2. Draw the axis of symmetry (x = 1).
3. Plot the x-intercepts (1 + √2/2, 0) and (1 - √2/2, 0).
4. Plot the y-intercept (0, 1).

Now, draw a smooth curve through these points to form the parabola.

Example 3: Graphing the Quadratic Function

Let’s graph the function f(x) = 2x² - 4x + 1:

1. Vertex: (1, -1)
2. Axis of symmetry: x = 1
3. x-intercepts: (1 + √2/2, 0) and (1 - √2/2, 0)
4. y-intercept: (0, 1)

Using this information, you can sketch the parabola accurately.

Common Mistakes

Summary

• The vertex of a quadratic function is found using h = -b/(2a) and k = f(h).
• The axis of symmetry is x = h.
• The x-intercepts are found by solving ax² + bx + c = 0 using the quadratic formula.
• The y-intercept is found by evaluating f(0) = c.
• Graph the function by plotting the vertex, intercepts, and drawing a smooth curve.

Check your understanding

1. What is the vertex of the quadratic function f(x) = -3x² + 6x + 2?
2. How do you find the axis of symmetry for a quadratic function?
3. What does it mean if the discriminant is negative?
4. Sketch the graph of the function f(x) = x² - 4x + 3, including the vertex and intercepts.