Graphing Quadratic Functions and Equations

Quadratic Functions and Their Graphs

Understanding Quadratic Functions

Quadratic functions are a fundamental topic in algebra and precalculus, forming the basis for many applications in trigonometry and higher mathematics. The general form of a quadratic function is:

• Standard Form: $y = ax^2 + bx + c$

• Vertex Form: $y = a(x-h)^2 + k$

Where a, b, and c are constants, and the graph of a quadratic function is a parabola.

Key Features of Parabolas

• Vertex: The highest or lowest point on the graph, given by $(h, k)$ in vertex form.

• Axis of Symmetry: The vertical line that passes through the vertex, given by $x = h$.

• Direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

• Y-intercept: The point where the graph crosses the y-axis, found by evaluating $y$ when $x = 0$.

Graphing Quadratic Functions

1. Identify the coefficients $a$, $b$, and $c$.

2. Find the vertex using $x = -\frac{b}{2a}$ and substitute back to find $y$.

3. Plot the vertex and the y-intercept.

4. Draw the axis of symmetry.

5. Plot additional points as needed and sketch the parabola.

Example

Given $y = x^2 - 4x + 3$:

• Find the vertex: $x = -\frac{-4}{2 \times 1} = 2$

• Substitute $x = 2$ into the equation: $y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$

• Vertex: $(2, -1)$

• Y-intercept: Set $x = 0$, $y = 0^2 - 4(0) + 3 = 3$

Practice Problems

1. Graph $y = x^2 + 2x - 3$ and identify the vertex and axis of symmetry.

2. Find the y-intercept and vertex for $y = -2x^2 + 4x + 1$.

Additional info:

These concepts are foundational for understanding more advanced topics such as inequalities, conic sections, and trigonometric applications involving quadratic equations.