GED Math - S04.4V3
Functions > Quadratic Functions
Important Instructions
"Before starting the test, let's review the essential concepts of Quadratic Functions. Please read each question carefully and respond to all 22 questions within TIME LIMIT of 25 min.
Understanding Quadratic Functions: Definition, Key Features, and Graph Interpretation
1. Definition of Quadratic Functions
Definition: A quadratic function is a polynomial function of degree 2, expressed in the standard form:
\( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants and \( a \neq 0 \).
- \( a \): Determines the direction and width of the parabola.
- \( b \): Influences the position of the vertex along the x-axis.
- \( c \): The y-intercept (\( f(0) = c \)).
2. The Graph of a Quadratic Function
The graph of a quadratic function is a U-shaped curve called a parabola. It has the following characteristics:
- Direction:
- If \( a > 0 \), the parabola opens upward.
- If \( a < 0 \), the parabola opens downward.
- Vertex: The highest or lowest point of the parabola, depending on its direction.
- Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves.
- X-Intercepts (Roots): The points where the graph crosses the x-axis (\( f(x) = 0 \)).
- Y-Intercept: The point where the graph crosses the y-axis (\( f(0) = c \)).
3. Vertex and Axis of Symmetry
- Vertex: The vertex of the parabola can be found using:
\( x = -\frac{b}{2a} \), and \( y = f\left(-\frac{b}{2a}\right) \).
Thus, the vertex is \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \).
- Axis of Symmetry: The equation of the axis of symmetry is:
\( x = -\frac{b}{2a} \).
4. Forms of Quadratic Functions
Quadratic functions can be written in different forms:
- Standard Form: \( f(x) = ax^2 + bx + c \).
- Vertex Form: \( f(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex.
- Factored Form: \( f(x) = a(x-p)(x-q) \), where \( p \) and \( q \) are the roots of the equation.
Example: Convert \( f(x) = 2x^2 + 8x + 6 \) to vertex form:
- Complete the square: \( f(x) = 2(x^2 + 4x) + 6 \).
- \( f(x) = 2((x+2)^2 - 4) + 6 = 2(x+2)^2 - 8 + 6 = 2(x+2)^2 - 2 \).
- Vertex form: \( f(x) = 2(x+2)^2 - 2 \), with vertex \( (-2, -2) \).
5. Finding X-Intercepts (Roots)
The roots of a quadratic function can be found by solving \( ax^2 + bx + c = 0 \) using:
- Factoring: If the equation can be factored easily.
- Quadratic Formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
- Completing the Square: Rearrange into a perfect square trinomial.
Example: Solve \( 2x^2 + 4x - 6 = 0 \) using the quadratic formula:
- \( a = 2, b = 4, c = -6 \).
- \( x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-6)}}{2(2)} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4} \).
- Roots: \( x = 1 \) and \( x = -2 \).
6. Applications of Quadratic Functions
- Modeling projectile motion in physics (\( h(t) = -16t^2 + vt + h_0 \)).
- Optimizing areas and profits in business.
- Solving problems involving speed, distance, and acceleration.
Final Tip
Mastery Tip: Understand the relationships between the forms of quadratic functions and their graphs. Practice converting between forms and interpreting key features like the vertex, intercepts, and axis of symmetry to solve real-world problems effectively.
