Which function is graphed

We will see these toolkit functions, combinations of toolkit functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties.

The graphs and sample table values are included with each function shown below. Improve this page Learn More. Skip to main content. Module 5: Function Basics.

Search for:. Identify Functions Using Graphs Learning Outcomes Verify a function using the vertical line test Verify a one-to-one function with the horizontal line test Identify the graphs of the toolkit functions.

How To: Given a graph, use the vertical line test to determine if the graph represents a function. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. If there is any such line, the graph does not represent a function. If no vertical line can intersect the curve more than once, the graph does represent a function. Let f be the function whose graph is drawn below.

Note that f a is not the smallest function value, f c is. However, if we consider only the portion of the graph in the circle above a, then f a is the smallest second coordinate.

Look at the circle on the graph above b. While f b is not the largest function value this function does not have a largest value , if we look only at the portion of the graph in the circle, then the point b, f b is above all the other points. So, f b is a relative maximum of f.

Indeed, f c is the absolute minimum of f, but it is also one of the relative minima. Here again we are giving definitions that appeal to your geometric intuition. The precise definitions are given in your text.

Finding the exact location of a function's relative extrema generally requires calculus. However, graphing utilities such as the Java Grapher may be used to approximate these numbers.

Suppose a is a number such that f a is a relative minimum. In applications, it is often more important to know where the function attains its relative minimum than it is to know what the relative minimum is. We will call the point 2,0 a relative minimum point.

In general, a relative extreme point is a point on the graph of f whose second coordinate is a relative extreme value of f. When you display the graph of f in the default viewing rectangle you see that f has one relative maximum point near -1,4 and one relative minimum point near 2, The approximations -1,4 and 2,-8 are not very close to the real relative extreme points, so we will use the zoom and trace features to improve the approximations.

When you click the Trace button, a point on the graph of f is indicated with a small circle. The coordinates of that point are reported in the two text boxes near the Trace button. If you select a larger Step Size from the pull down menu, then the trace point moves farther with each click. It is possible to move faster with the enter key than with the mouse. A function is a set of rules so that for every input we get only one output.

We represent functions in math as equations with two variables: x and y. So for every x we plug into the equation, we only get one y. We can represent functions on a graph using the coordinate plane with our coordinates being x the input and y the output in the same way as we graphed things before, with our point x, y.

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Describing the Graph of a Function Sometimes you need to describe the graph of a function in a non-symbolic way. For example, you may be asked whether a function is increasing or decreasing; whether it has one minimum value or maximum value, or several such values whether it is linear or not whether the rate of change is constant, increasing, or decreasing whether it has an upper or lower bound.