In Exercises 39–50, graph the given functions, f and g, in the same rectangular coordinate system. Select integers for x, starting with -2 and ending with 2. Once you have obtained your graphs, describe how the graph of g is related to the graph of f. f(x) = x, g(x) = x + 3
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<Step 1: Identify the functions.> The functions given are f(x) = x and g(x) = x + 3. These are both linear functions, which means their graphs will be straight lines.
<Step 2: Create a table of values for f(x).> Choose integer values for x from -2 to 2. Calculate the corresponding y-values for f(x) = x. For example, if x = -2, then f(-2) = -2.
<Step 3: Create a table of values for g(x).> Using the same x-values from -2 to 2, calculate the corresponding y-values for g(x) = x + 3. For example, if x = -2, then g(-2) = -2 + 3 = 1.
<Step 4: Plot the points on a graph.> Use the tables of values to plot the points for both f(x) and g(x) on the same rectangular coordinate system. Connect the points for each function to form straight lines.
<Step 5: Analyze the relationship between the graphs.> Observe how the graph of g(x) = x + 3 is related to the graph of f(x) = x. Notice that g(x) is a vertical shift of f(x) by 3 units upward. This means every point on the graph of f(x) is moved 3 units up to get the graph of g(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Graphing Functions
Graphing functions involves plotting points on a coordinate system where the x-axis represents the input values and the y-axis represents the output values. For the functions f(x) = x and g(x) = x + 3, you will calculate the corresponding y-values for selected x-values, which helps visualize the relationship between the two functions.
Transformation of functions refers to the changes made to the graph of a function based on modifications to its equation. In this case, g(x) = x + 3 represents a vertical shift of the graph of f(x) = x upwards by 3 units, illustrating how the output values of g are consistently 3 greater than those of f for the same input.
A coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). It allows for the representation of mathematical functions graphically. Understanding how to plot points and interpret the axes is crucial for analyzing the relationship between the graphs of f and g.