Parabolas

Parabolas

A parabola is the graph of a quadratic function. Quadratic functions can model various real-world scenarios, such as projectile motion. Understanding parabolas is important for solving problems in mathematics, physics, and engineering.

Understanding Parabolas

A parabola is defined by a quadratic function of the form:

f(x) = ax² + bx + c

where a, b, and c are constants and a ≠ 0. The graph of this function is a curve that opens either upwards or downwards depending on the sign of a:

• If a > 0, the parabola opens upwards.
• If a < 0, the parabola opens downwards.

Characteristics of Parabolas

Vertex

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. The vertex can be found using the formula:

Vertex: (h, k) where h = -b/(2a) and k = f(h).

Axis of Symmetry

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

x = h.

Intercepts

The y-intercept can be found by substituting x = 0 into the equation:

y-intercept: (0, c).

The x-intercepts can be found by solving the equation:

ax² + bx + c = 0.

Example: Finding the Vertex and Axis of Symmetry

Consider the quadratic function:

f(x) = 2x² - 4x + 1.

To find the vertex:

1. Calculate h: h = -(-4)/(2 * 2) = 1.
2. Calculate k: k = f(1) = 2(1)² - 4(1) + 1 = -1.

The vertex is (1, -1). The axis of symmetry is x = 1.

Sketching Parabolas

To sketch a parabola, follow these steps:

1. Identify the vertex and plot it on the graph.
2. Find the y-intercept and plot it.
3. Find the x-intercepts, if they exist, and plot them.
4. Draw the axis of symmetry.
5. Sketch the curve, ensuring it is symmetric about the axis of symmetry.

Example: Sketching a Parabola

Using the function from the previous example:

f(x) = 2x² - 4x + 1.

1. Vertex: (1, -1).

2. y-intercept: (0, 1).

3. x-intercepts: Solve 2x² - 4x + 1 = 0 using the quadratic formula:

x = [4 ± √((-4)² - 4 * 2 * 1)]/(2 * 2) = [4 ± √(16 - 8)]/4 = [4 ± √8]/4 = [4 ± 2√2]/4 = 1 ± √2/2.

4. The x-intercepts are approximately (2.414, 0) and (-0.414, 0).

Now plot these points and sketch the parabola.

Discriminant and Roots

The discriminant of a quadratic function helps determine the number of x-intercepts:

D = b² - 4ac.

Depending on the value of D:

• If D > 0, there are two distinct x-intercepts.
• If D = 0, there is one x-intercept (the vertex touches the x-axis).
• If D < 0, there are no x-intercepts.

Example: Using the Discriminant

For the function f(x) = 2x² - 4x + 1:

D = (-4)² - 4(2)(1) = 16 - 8 = 8 (D > 0, two x-intercepts).

Solving Quadratic Inequalities

To solve a quadratic inequality, you can sketch the corresponding parabola and determine the intervals where the function is above or below the x-axis:

For example, to solve:

2x² - 4x + 1 < 0.

1. Find the x-intercepts (as done earlier).

2. Sketch the parabola.

3. Identify the intervals where the parabola is below the x-axis.

Common Mistakes

Summary

• A parabola is the graph of a quadratic function.
• The vertex is the highest or lowest point on the graph.
• The axis of symmetry is a vertical line through the vertex.
• Intercepts can be found by substituting values into the quadratic equation.
• The discriminant determines the number of x-intercepts.

Check your understanding

1. What is the vertex of the parabola defined by f(x) = -3x² + 6x - 2?
2. How does the value of a affect the shape of the parabola?
3. What does the discriminant tell you about the roots of a quadratic equation?
4. How would you sketch the parabola defined by f(x) = x² - 4?