Quadratic functions
Mathematics - Grade 11 · Functions and Graphs
Quadratic Functions
A quadratic function is a polynomial function of degree 2. It can be expressed in the standard form:
f(x) = ax^2 + bx + c
where:
- a is the coefficient of x^2 (it cannot be zero),
- b is the coefficient of x,
- c is the constant term.
Graph of a Quadratic Function
The graph of a quadratic function is a parabola. It opens upwards if a > 0 and downwards if a < 0. The vertex of the parabola is its highest or lowest point, depending on the direction it opens.
Remember: The vertex form of a quadratic function is:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
Finding the Vertex
The vertex can be found using the formula:
x = -b/(2a)
Once you find the x-coordinate, substitute it back into the function to find the y-coordinate.
Example 1
Find the vertex of the quadratic function f(x) = 2x^2 - 4x + 1.
Step 1: Identify the coefficients:
- a = 2
- b = -4
- c = 1
Step 2: Use the vertex formula:
x = -(-4)/(2 × 2) = 4/4 = 1
Step 3: Substitute x back into the function to find y:
f(1) = 2(1)^2 - 4(1) + 1 = 2 - 4 + 1 = -1
Thus, the vertex is (1, -1).
Intercepts of the Quadratic Function
The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. These can be found by setting f(x) = 0 and solving for x:
ax^2 + bx + c = 0
This equation can be solved using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Example 2
Find the x-intercepts of the function f(x) = 2x^2 - 4x + 1.
Step 1: Identify the coefficients:
- a = 2
- b = -4
- c = 1
Step 2: Calculate the discriminant:
D = b^2 - 4ac = (-4)^2 - 4(2)(1) = 16 - 8 = 8
Step 3: Use the quadratic formula:
x = (4 ± √8) / (2 × 2)
Step 4: Simplify:
x = (4 ± 2√2) / 4 = 1 ± (√2)/2
The x-intercepts are:
x = 1 + (√2)/2 and x = 1 - (√2)/2.
Graphing the Quadratic Function
To graph a quadratic function, follow these steps:
- Find the vertex.
- Find the x-intercepts.
- Find the y-intercept by evaluating f(0).
- Plot the points on a Cartesian plane.
- Draw the parabola through the points.
Example 3
Graph the function f(x) = 2x^2 - 4x + 1.
Step 1: The vertex is (1, -1).
Step 2: The x-intercepts are x = 1 + (√2)/2 and x = 1 - (√2)/2.
Step 3: Find the y-intercept:
f(0) = 2(0)^2 - 4(0) + 1 = 1
The y-intercept is (0, 1).
Step 4: Plot the points (1, -1), (0, 1), and the x-intercepts.
Properties of Quadratic Functions
Quadratic functions have several important properties:
- The vertex indicates the maximum or minimum value of the function.
- The axis of symmetry is the vertical line that passes through the vertex, given by x = h.
- The direction of the parabola is determined by the sign of a.
- The y-intercept is found by evaluating f(0).
Tip: Always check the discriminant D = b^2 - 4ac to determine the nature of the roots:
- If D > 0, there are two distinct real roots.
- If D = 0, there is one real root (the vertex touches the x-axis).
- If D < 0, there are no real roots (the parabola does not intersect the x-axis).
Check your understanding
- Find the vertex of the quadratic function f(x) = -x^2 + 4x - 3.
- Determine the x-intercepts of the function f(x) = x^2 - 2x - 3.
- Sketch the graph of the function f(x) = 3x^2 + 6x + 2, indicating the vertex and intercepts.
- What can you conclude about the number of real roots of the function f(x) = 2x^2 + 4x + 5 based on its discriminant?