- Identify key characteristics of parabolas from the graph.
- Understand how the graph of a parabola is related to its quadratic function.
- Draw the graph of a quadratic function.
- Solve problems involving a quadratic function’s minimum or maximum value.
Key Features of a Parabola
The Main Idea
The graph of a quadratic function is a U-shaped curve called a parabola.
Every parabola has these important features:
- Vertex: The turning point—either the highest or lowest point on the graph
- Axis of symmetry: A vertical line through the vertex that divides the parabola into two mirror images
- y-intercept: Where the parabola crosses the y-axis
- x-intercepts (zeros/roots): Where the parabola crosses the x-axis (if it does)
- Direction of opening: Whether the parabola opens upward or downward
Domain and Range of Quadratic Functions
The Main Idea
For quadratic functions:
- The domain is always all real numbers – you can input any value for x
- The range depends on the vertex and whether the parabola opens up or down
If the parabola opens upward ([latex]a > 0[/latex]):
- The vertex is the minimum point
- Range: [latex]y \geq k[/latex] or [latex][k, \infty)[/latex] where [latex]k[/latex] is the y-coordinate of the vertex
If the parabola opens downward ([latex]a < 0[/latex]):
- The vertex is the maximum point
- Range: [latex]y \leq k[/latex] or [latex](-\infty, k][/latex] where [latex]k[/latex] is the y-coordinate of the vertex
Question Help: Finding Domain and Range
- The domain is always all real numbers: [latex](-\infty, \infty)[/latex].
- Determine whether [latex]a[/latex] is positive or negative.
- If [latex]a > 0[/latex], the parabola opens upward and has a minimum at the vertex.
- If [latex]a < 0[/latex], the parabola opens downward and has a maximum at the vertex.
- Find the y-coordinate of the vertex, [latex]k[/latex].
- Write the range:
- Minimum: [latex][k, \infty)[/latex]
- Maximum: [latex](-\infty, k][/latex]
You can view the transcript for “Finding Domain & Range – with Parabolas” here (opens in new window).
Forms of Quadratic Functions
Quadratic functions can be written in different forms, each useful for different purposes:
General Form: [latex]f(x) = ax^2 + bx + c[/latex]
- Shows the y-intercept directly: [latex]c[/latex]
- Useful for identifying basic properties
Standard Form (Vertex Form): [latex]f(x) = a(x - h)^2 + k[/latex]
- Shows the vertex directly: [latex](h, k)[/latex]
- Best for graphing and transformations
Key properties from the equation:
- If [latex]a > 0[/latex], parabola opens upward
- If [latex]a < 0[/latex], parabola opens downward
- The axis of symmetry is [latex]x = -\frac{b}{2a}[/latex] (general form) or [latex]x = h[/latex] (standard form)
Finding the Vertex
When a quadratic is in general form [latex]f(x) = ax^2 + bx + c[/latex], you can find the vertex using these formulas:
x-coordinate of vertex: [latex]h = -\frac{b}{2a}[/latex]
You can view the transcript for “Find the Vertex of the Parabola in Standard Form-Example 1” here (opens in new window).
Transformations of Quadratic Functions
The Main Idea
Understanding how changes to the equation affect the graph helps you sketch parabolas quickly without making tables.
Standard Form: [latex]f(x) = a(x - h)^2 + k[/latex]
- Changing [latex]k[/latex]: Shifts the parabola vertically
- [latex]k > 0[/latex]: shifts up
- [latex]k < 0[/latex]: shifts down
- Changing [latex]h[/latex]: Shifts the parabola horizontally
- [latex]h > 0[/latex]: shifts right
- [latex]h < 0[/latex]: shifts left
- Remember: [latex](x - h)[/latex] means opposite direction!
- Changing [latex]a[/latex]: Affects width and direction
- [latex]|a| > 1[/latex]: narrower (vertical stretch)
- [latex]0 < |a| < 1[/latex]: wider (vertical compression)
- [latex]a < 0[/latex]: opens downward (reflection over x-axis)
You can view the transcript for “Graphing Quadratic Functions Using Transformations” here (opens in new window).