Look at the coefficient for x 2 , not the coefficient for x. The correct answer is Down, because a the coefficient of x 2 is negative. Radical Functions. You can also graph radical functions such as square root functions by choosing values for x and finding points that will be on the graph. Think about the basic square root function,.
Notice that all the values for x in the table are perfect squares. Since you are taking the square root of x , using perfect squares makes more sense than just finding the square roots of 0, 1, 2, 3, 4, etc. Recall that x can never be negative here because the square root of a negative number would be imaginary, and imaginary numbers cannot be graphed. There are also no values for x that will result in y being a negative number. Take a look at the graph. As with parabolas, multiplying and adding numbers makes some changes, but the basic shape is still the same.
Here are some examples. Multiplying by a positive value changes the width of the half-parabola. Multiplying by a negative number gives you the other half of a horizontal parabola. Adding a value outside the radical moves the graph up or down.
Think about it as adding the value to the basic y value of , so a positive value added moves the graph up. Adding a value inside the radical moves the graph left or right.
Think about it as adding a value to x before you take the square root—so the y value gets moved to a different x value.
Notice that as x gets greater, adding or subtracting a number inside the square root has less of an effect on the value of y! Before making a table of values, look at the function equation to get a general idea what the graph should look like. Choose values that will make your calculations easy. You want x — 1 to be a perfect square 0, 1, 4, 9, and so on so you can take the square root. There is no need to choose x values less than 1 for your table! Which of the following is a graph of?
Recall that the graph of is entirely under the x -axis, in Quadrant IV. The graph of will be similar but wider. The correct answer is Graph C. Let be a function, then. The reciprocal function is symmetric along the origin, but it never touches the origin itself. The properties of a reciprocal function is given below. Let , be the step function where. If you remember these basic graph of functions used in algebra, then it is easier to learn higher and complex graphs.
Later , when you learn calculus, visualizing concepts is much easier with a graph of function. All the points on a vertical line have the same first coordinate, so if a vertical line hits a graph twice, then there are two points on the graph with the same first coordinate. If that happens, the graph is not the graph of a function. Think of a point moving on the graph of f. As the point moves toward the right it rises. This is what it means for a function to be increasing. Your text has a more precise definition, but this is the basic idea.
The function f above is increasing everywhere. In general, there are intervals where a function is increasing and intervals where it is decreasing. The function graphed above is decreasing for x between -3 and 2. It is increasing for x less than -3 and for x greater than 2.
Some of the most characteristics of a function are its Relative Extreme Values. Points on the functions graph corresponding to relative extreme values are turning points, or points where the function changes from decreasing to increasing or vice versa. 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. How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.
Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. If there is any such line, the function is not one-to-one. If no horizontal line can intersect the curve more than once, the function is one-to-one. Example: Applying the Horizontal Line Test Consider the functions a , and b shown in the graphs below. Show Solution The function in a is not one-to-one.
Try It. Try It In this exercise, you will graph the toolkit functions using an online graphing tool. Graph each toolkit function using function notation. Make a table of values that references the function and includes at least the interval [-5,5]. Did you have an idea for improving this content? Licenses and Attributions.
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