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

Verified step by step guidance

1. Identify the given functions: \(f(x) = x\) and \(g(x) = x + 3\).
2. Create a table of values for \(x\) starting from \(-2\) to \(2\). For each \(x\), calculate \(f(x)\) and \(g(x)\) separately.
3. For \(f(x) = x\), the values will be the same as \(x\) itself. For \(g(x) = x + 3\), add 3 to each \(x\) value to find the corresponding \(g(x)\) values.
4. Plot the points for both functions on the same coordinate system using the calculated values. For example, plot \((x, f(x))\) and \((x, g(x))\) for each \(x\) from \(-2\) to \(2\).
5. Observe the graphs: describe how the graph of \(g\) is related to the graph of \(f\). Consider how adding 3 to \(x\) affects the position of the graph compared to \(f(x)\).

Key Concepts

Function Notation and Evaluation

Function notation, such as f(x) and g(x), represents a rule that assigns each input x to exactly one output. Evaluating a function means substituting a specific value for x to find the corresponding output, which is essential for plotting points on a graph.

Graphing Linear Functions

Linear functions like f(x) = x produce straight-line graphs. Plotting points for selected x-values and connecting them helps visualize the function. Understanding the slope and intercept aids in sketching and comparing graphs.

Vertical Translation of Graphs

Adding a constant to a function, as in g(x) = x + 3, shifts the graph vertically. Specifically, g(x) is the graph of f(x) moved up by 3 units. Recognizing this transformation helps describe the relationship between the two graphs.