The graph of f(x)=5 x follow. Use this graph to match each...

The graph of $f(x)=5 x$ follow. Use this graph to match each linear function with its graph. See Examples 1 and 2.
Graph can't copy
(a) Graph can't copy
(b) Graph can't copy
(c) Graph can't copy
(d) Graph can't copy

$f(x)=5 x-2$

Key Concepts

Linear Functions

Linear functions are mathematical expressions that produce straight-line graphs. They are generally written in the form f(x) = mx + b, where m represents the slope and b represents the y-intercept. This form clearly indicates both the steepness and the vertical position of the graph on the coordinate plane.

Slope

The slope of a linear function is the measure of its steepness, expressed as the ratio of the rise (vertical change) to the run (horizontal change) between any two points on the line. A constant slope confirms that the function is linear, meaning it increases or decreases at a consistent rate throughout its domain.

y-intercept

The y-intercept is the point where the graph of a function crosses the y-axis, corresponding to the function value when x = 0. In the slope-intercept form f(x) = mx + b, the y-intercept is given by the constant b and serves as the starting position of the graph before any additional transformations are applied.

Vertical Shifts

Vertical shifts occur when a constant is added to or subtracted from a function, which moves the graph up or down on the coordinate plane without affecting its shape or slope. For example, converting the function f(x) = 5x to f(x) = 5x - 2 shifts the entire graph downward by 2 units.

Graph Transformations

Graph transformations involve applying operations such as translations (shifts), scaling (stretches or compressions), and reflections to basic functions in order to obtain new graphs. Recognizing these transformations is crucial when matching a modified function to its corresponding graph, as it's important to identify changes in position, slope, or other graphical features relative to a known graph.

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