Interpreting Key Features of Quadratic Functions

1. Interpreting Key Features of Quadratic Functions Adapted from Walch Education

2. Key Features • The ordered pair that corresponds to an x-intercept is always of the form (x, 0). The x-intercepts are also the solutions of a quadratic function. • The ordered pair that corresponds to a y-intercept is always of the form (0, y). • The vertex is the point on a parabola where the graph changes direction. • The maximum or minimum of the function occurs at the vertex of the parabola. 5.5.1: Interpreting Key Features of Quadratic Functions

3. Key Features, continued • The vertex is also the point where the parabola changes from increasing to decreasing. • Increasing refers to the interval of a function for which the output values are becoming larger as the input values are becoming larger. • Decreasing refers to the interval of a function for which the output values are becoming smaller as the input values are becoming larger. 5.5.1: Interpreting Key Features of Quadratic Functions

4. Key Features, continued • Any point to the right or left of the parabola is equidistant to another point on the other side of the parabola. • A parabola only increases or decreases as x becomes larger or smaller. • Read the graph from left to right to determine when the function is increasing or decreasing. • Trace the path of the graph with a pencil tip. If your pencil tip goes down as you move toward increasing values of x, then f(x) is decreasing. 5.5.1: Interpreting Key Features of Quadratic Functions

5. Key Features, continued • If your pencil tip goes up as you move toward increasing values of x, then f(x) is increasing. • For a quadratic, if the graph has a minimum value, then the quadratic will start by decreasing toward the vertex, and then it will increase. • If the graph has a maximum value, then the quadratic will start by increasing toward the vertex, and then it will decrease. • The vertex is called an extremum. Extrema are the maxima or minima of a function. 5.5.1: Interpreting Key Features of Quadratic Functions

6. Key Features, continued • The concavity of a parabola is the property of being arched upward or downward. • A quadratic with positive concavity will increase on either side of the vertex, meaning that the vertex is the minimum or lowest point of the curve. • A quadratic with negative concavity will decrease on either side of the vertex, meaning that the vertex is the maximum or highest point of the curve. 5.5.1: Interpreting Key Features of Quadratic Functions

7. Key Features, continued • A quadratic that has a minimum value is concave up because the graph of the function is bent upward. • A quadratic that has a maximum value is concave down because the graph of the function is bent downward. 5.5.1: Interpreting Key Features of Quadratic Functions

8. Inflection Point • The inflection point of a graph is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. • The vertex of a quadratic function is also the point of inflection. 5.5.1: Interpreting Key Features of Quadratic Functions

9. Inflection Point 5.5.1: Interpreting Key Features of Quadratic Functions

10. End Behavior • End behavior is the behavior of the graph as x becomes larger or smaller. • If the highest exponent of a function is even, and the coefficient of the same term is positive, then the function is approaching positive infinity as x approaches both positive and negative infinity. • If the highest exponent of a function is even, but the coefficient of the same term is negative, then the function is approaching negative infinity as x approaches both positive and negative infinity. 5.5.1: Interpreting Key Features of Quadratic Functions

11. Even/Odd Function Functions can be defined as odd or even based on the output yielded when evaluating the function for –x. • For an odd function, f(–x) = –f(x). That is, if you evaluate a function for –x, the resulting function is the opposite of the original function. • For an even function, f(–x) = f(x). That is, if you evaluate a function for –x, the resulting function is the same as the original function. • If evaluating the function for –x does not result in the opposite of the original function or the original function, then the function is neither odd nor even. 5.5.1: Interpreting Key Features of Quadratic Functions

12. Ms. Dambreville Thanks for Watching!