Interpreting Key Features of Quadratic Functions

Adapted from Walch EducationInterpreting Key Features of Quadratic Functions

5.5.1: Interpreting Key Features of Quadratic Functions2Key FeaturesThe 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 Functions3Key Features, continuedThe 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 Functions4Key Features, continuedAny 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 Functions5Key Features, continuedIf 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 Functions6Key Features, continuedThe 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 Functions7Key Features, continuedA 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 Functions8Inflection PointThe 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 Functions9Inflection Point

5.5.1: Interpreting Key Features of Quadratic Functions10End BehaviorEnd 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 Functions11Even/Odd FunctionFunctions 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.

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