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The x-intercept of a function is the point where the graph of the function crosses the x-axis. In simpler terms, it's where the y-value equals zero. To find the x-intercept for a linear equation like the one we have here, which is given as: \[f(x) = -5x + 15\]You need to solve the equation for x when we set the function to zero:

• Set the function equal to zero: \[0 = -5x + 15\]
• Add 5x to each side to get: \[5x = 15\]
• Divide both sides by 5 to solve for x: \[x = 3\]

Therefore, the x-intercept is at (3, 0). This means if you were to draw the line of the graph, it would cross the x-axis at the point (3, 0). Knowing how to find the x-intercept is crucial for graphing linear functions because it provides one of the two main points needed to draw the line. Additionally, intercepts help in understanding where the line intersects the axes of the graph, which is fundamental in analyzing the behavior of linear equations.

The y-intercept is another fundamental component of graphing linear equations. It's the point where the graph intersects the y-axis, meaning it happens when the x-value is zero. For the given linear function \[f(x) = -5x + 15\], we can easily find this point by:

• Substituting \(x = 0\) into the function: \[f(0) = -5(0) + 15\]
• This simplifies to: \[15\]

Thus, the y-intercept is at (0, 15). It's the point where our graph of the linear equation will touch or cross the y-axis.Understanding the y-intercept is important because it, along with the x-intercept, provides vital information for quickly sketching the graph of a linear equation. It offers insight into the starting point of the graph when x equals zero. Furthermore, the coordinate (0, b) in the slope-intercept form \(y = mx + b\) directly reflects the y-intercept. In this instance, the graph starts at 15 on the y-axis, indicating the overall vertical shift of the line with respect to the origin.

Linear equations represent a straight line on a graph and are a foundational concept in algebra. A common form of a linear equation is the slope-intercept form, expressed as:\[y = mx + b\]Where \(m\) represents the slope, and \(b\) is the y-intercept. For our function:

• The equation is \(f(x) = -5x + 15\).
• Here, the slope \(m\) is -5.
• The y-intercept \(b\) is 15.

The slope \(m\) tells us how steep the line is and the direction in which the line goes. A negative slope, like -5, indicates the line goes down as it moves from left to right. Linear equations like this one are simple yet powerful tools for modeling relationships where one quantity changes consistently in relation to another. They make it easy to predict one variable when the other is known, whether you're calculating costs, distances, or other variables that follow a linear pattern. Graphing linear equations involves understanding both the intercepts and the slope, enabling you to plot a line that fits given data or conditions beautifully.