How To Choose The Function Whose Graph Is Given Below

April 30, 2024 Posting Komentar

Choose the function whose graph is given below. Y = csc x Y= tan x Y from brainly.com

As we continue to learn about mathematics, one of the most important topics that we encounter is functions. A function can be defined as a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. One way that these functions can be represented is through their graphs. In this article, we will discuss how to choose the function whose graph is given below.

Understanding the Graph of a Function

Before we proceed with our discussion, let us first understand what a graph of a function is. A graph is a visual representation of how the input and output of a function are related. It is usually plotted on a Cartesian plane, where the horizontal axis represents the input values and the vertical axis represents the output values.

Each point on the graph represents a pair of input and output values. The shape and position of the graph can help us determine the behavior of the function. For instance, a graph that is increasing from left to right indicates that the function is also increasing as the input values increase.

Determining the Type of Function

One of the first things that we need to do when choosing the function whose graph is given below is to determine the type of function. There are different types of functions that we encounter in mathematics, such as linear, quadratic, exponential, and trigonometric functions, among others.

To determine the type of function, we need to look at the shape of the graph. A linear function has a straight line graph, a quadratic function has a parabolic graph, an exponential function has an increasing or decreasing curve, and a trigonometric function has a periodic wave-like graph.

Determining the Domain and Range

After determining the type of function, we need to determine its domain and range. The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

To determine the domain, we need to look at the input values that are allowed in the graph. For instance, if the graph is a straight line, then all real numbers are allowed as input values. However, if the graph has a hole or a vertical asymptote, then there are restrictions on the input values.

To determine the range, we need to look at the output values that are possible in the graph. For instance, if the graph is a parabola that opens upward, then the range is all real numbers greater than or equal to the vertex of the parabola. If the graph is an exponential curve, then the range is all positive real numbers.

Finding the Equation of the Function

Once we have determined the type of function, domain, and range, we can proceed to find its equation. The equation of a function is an algebraic expression that relates the input and output values of the function.

For instance, the equation of a linear function can be written in the form y = mx + b, where m is the slope of the line and b is the y-intercept. The equation of a quadratic function can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. The equation of an exponential function can be written in the form y = ab^x, where a and b are constants.

Checking for Accuracy

After finding the equation of the function, we need to check if it is accurate. One way to do this is by plugging in some input values and checking if the output values match the ones in the graph.

For instance, if the graph is a straight line passing through the points (1, 3) and (4, 9), then we can check if the equation y = 2x + 1 matches the input and output values of the function. If we plug in x = 1, we get y = 3, and if we plug in x = 4, we get y = 9, which matches the points on the graph.

Common Mistakes to Avoid

When choosing the function whose graph is given below, there are some common mistakes that we need to avoid. One of these is assuming that the graph represents a certain type of function without thoroughly examining its shape and behavior.

Another mistake is assuming that the domain and range of the function are unrestricted without looking at the potential restrictions on its input and output values. Moreover, we need to be careful when finding the equation of the function and double-check if it matches the input and output values in the graph.

Examples

Let us now look at some examples to illustrate how to choose the function whose graph is given below.

Example 1

The graph below represents a linear function. We can see that it passes through the points (1, 3) and (4, 9).

To determine the equation of the function, we can use the slope-intercept form y = mx + b. We can find the slope by using the formula (y2 - y1) / (x2 - x1), where (x1, y1) = (1, 3) and (x2, y2) = (4, 9).

Slope = (9 - 3) / (4 - 1) = 2

Now, we can plug in the slope and one of the points to find the y-intercept.

y = mx + b

3 = 2(1) + b

b = 1

Therefore, the equation of the function is y = 2x + 1. We can check if it matches the input and output values of the function.

If we plug in x = 1, we get y = 3, and if we plug in x = 4, we get y = 9, which matches the points on the graph.

Example 2

The graph below represents a quadratic function. We can see that it has a vertex at (2, -1) and passes through the point (1, 0).

To determine the equation of the function, we can use the vertex form y = a(x - h)^2 + k, where (h, k) is the vertex.

y = a(x - 2)^2 - 1

Now, we can plug in the point (1, 0) to find the value of a.

0 = a(1 - 2)^2 - 1

a = 1

Therefore, the equation of the function is y = (x - 2)^2 - 1. We can check if it matches the input and output values of the function.

If we plug in x = 2, we get y = -1, which matches the vertex of the graph. If we plug in x = 1, we get y = 0, which matches the point on the graph.

Conclusion

Choosing the function whose graph is given below can be a challenging task, but with the right approach and understanding, we can determine the type of function, domain, range, and equation accurately. We need to be careful when examining the shape and behavior of the graph, as well as when finding the equation of the function. By avoiding common mistakes and checking for accuracy, we can confidently choose the function that represents the given graph.

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