Overview
This lecture explains the quadratic function, its properties, key points on its graph, and step-by-step instructions on how to graph it accurately.
What is a Quadratic Function?
- A function relates each value of ( x ) (input) to a single value of ( f(x) ) (output).
- A quadratic function has the form ( f(x) = ax^2 + bx + c ), with ( a \neq 0 ).
- ( a ), ( b ), and ( c ) are real number coefficients; ( a ) cannot be zero or it would be linear, not quadratic.
Key Properties of Quadratic Functions
- The highest degree of ( x ) is 2; this defines the function as quadratic (second degree).
- The graph of a quadratic function is a parabola.
- If ( a > 0 ), the parabola opens upward ("smile"); if ( a < 0 ), it opens downward ("frown").
- The greater the absolute value of ( a ), the narrower the parabola; the smaller, the wider.
- The parabola is symmetric about a vertical line called the axis of symmetry.
Graph Features and Calculations
- The vertex is the maximum or minimum point and is found at ( x = \frac{-b}{2a} ).
- Substitute this ( x ) into the function to find the vertex's ( y )-value.
- The axis of symmetry is vertical at the ( x )-coordinate of the vertex.
- The intersection with the ( y )-axis is found by plugging ( x = 0 ) into the function.
- Intersections with the ( x )-axis (the roots) are found by setting ( f(x) = 0 ) and using the quadratic formula.
Domain, Range (Image), and Behavior
- The domain of a quadratic function is all real numbers (( -\infty, \infty )).
- The range (image) depends on the vertex: for ( a > 0 ), it starts at the vertex's ( y )-value and increases; for ( a < 0 ), it decreases from the vertex.
- The function increases or decreases based on interval partitions determined by the vertex.
Steps to Graph a Quadratic Function
- Identify ( a ), ( b ), and ( c ) in the given quadratic function.
- Find the vertex and axis of symmetry.
- Determine ( x )- and ( y )-intercepts.
- Create a value table to plot additional points for accuracy.
- Draw the symmetric parabola through all calculated points.
Key Terms & Definitions
- Quadratic Function — A function with the highest power of ( x ) as 2, in the form ( ax^2 + bx + c ).
- Vertex — The highest or lowest point of the parabola.
- Axis of Symmetry — Vertical line dividing the parabola symmetrically, ( x = \frac{-b}{2a} ).
- Domain — All possible input values (( x )), usually all real numbers.
- Range (Image) — All possible output values (( f(x) )), starts/ends at the vertex depending on ( a ).
- Roots/Zeros — ( x )-values where the parabola crosses the ( x )-axis (( f(x)=0 )).
Action Items / Next Steps
- Practice graphing several quadratic functions using the outlined steps.
- Prepare a table of values and plot key points for each function assigned as homework.