The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
A parabola either opens upward or downward. This is called its direction.
At the vertex, the function changes from increasing to decreasing, or vice versa.
All parabolas are symmetric, meaning there exists a line that divides the graph into two mirror images. For quadratic functions, that line is always parallel to the -axis, and is called the axis of symmetry.
Depending on its rule, a parabola can intersect the -axis at or points. Since the function's value at an -intercept is always these points are called zeros, or sometimes roots.
When a quadratic function is written in standard form, it's possible to use and to determine characteristics of its graph.
The direction of the graph is determined by the sign of To understand why, consider the quadratic function Since all squares are positive, will always be positive. When is positive, then is also positive. Thus, when moving away from the origin in either direction, the graph extends upward. Similarly, when is negative, will be negative. Thus, the graph will extend downward for all -values.
The -intercept of a quadratic function is given by specifically at This is because substituting into standard form yields the following.
Vertex form is an algebraic format used to express quadratic function rules.
From the graph, we can connect the following characterisitcs to the function rule.Notice that although the factor in the function rule shows is actually equal to This coincides with a horizontal translation of a quadratic function.
Quadratic function rules can be expressed in factored form.
As is the case with standard form and vertex form, gives the direction of the parabola. When the parabola faces upward, and when it faces downward. Additionally, the zeros of the parabola lie at Because the points of a parabola with the same -coordinate are equidistant from the axis of symmetry, the axis of symmetry lies halfway between the zeros. Consider the graph of
From the graph, the following characteristics can be connected to the function rule.
Determine the characteristics of the quadratic functions
For the following functions, determine the direction, the axis of symmetry, vertex, and zeros.
We'll focus on each function individually, starting with
A function's rate of change gives an indication of how its outputs () change with respect to its inputs ().
When the rate of change is constant, the function is linear. However, when the function is not linear, it's possible to determine an average rate of change over an arbitrary interval. This is an increase or decrease between the endpoints of the interval. If the -coordinates of the endpoints are and and the function is the average rate of change is defined as follows.
Determine the average rate of change
The function is quadratic.
Determine the average rate of change over the interval
To determine the average rate of change, we must know the -values of the endpoints. Since the interval is the -values are and Let's mark these points on the graph to find their corresponding function values.