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In a linear function, the slope is a key concept that dictates the steepness of the line. If we look at the formula \( f(x) = mx + b \), the slope is represented by \( m \). It describes how much \( y \) changes for a unit increase in \( x \). The slope can be thought of as "rise over run." This means it represents two points on the line:

• Rise: The vertical change
• Run: The horizontal change

In a real-world context, the slope helps us understand rates of change, such as speed or cost per item. For example, if you have a slope of 5, for every unit you move right (along the x-axis), your line moves up 5 units (along the y-axis). To find the slope from the equation, identify the coefficient of \( x \), which is \( m \). This number tells you immediately about the direction and steepness of the graph, without plotting it.

The y-intercept is another important feature of a linear function. It tells us where the line crosses the y-axis. In the formula \( f(x) = mx + b \), the y-intercept is represented by \( b \). To find it, set \( x = 0 \) in the equation, and solve for \( f(x) \). The y-intercept is the value of \( f(x) \) when \( x \) is zero. This point provides valuable information about the starting position of the line on the graph:

• If \( b = 0 \), the line passes through the origin.
• If \( b > 0 \), the line starts above the origin.
• If \( b < 0 \), the line starts below the origin.

Understanding the y-intercept helps in visualizing the graph and knowing its initial position relative to the axes without drawing the line.

Evaluating a function involves finding the value of the function at a particular point. Given \( f(x) = mx + b \), to evaluate the function at \( x + h \), substitute \( x + h \) into the equation. This gives:

• Replace \( x \) with \( x + h \)
• Simplify the expression: \( f(x+h) = m(x+h) + b \)
• This expands to \( f(x+h) = mx + mh + b \)

By evaluating at a new point, such as \( x + h \), you can understand how shifts in the \( x \) value will impact the \( f(x) \) value. This process is essential not only for verifying algebraic properties of functions but also in real-world applications where you calculate changes or predict outcomes.

True or false questions require us to verify a statement against established rules or calculations. In the exercise, the statement \( f(x+h) = f(x) + mh \) needed to be verified. We evaluated the function at \( x + h \) and simplified it to check the given relation. This involves:

• Substituting within the function
• Simplifying to see if both sides of the statement match
• Comparing it directly to the given statement

Once the verification is complete, you can confidently mark the statement as true or false. This verification process, much like a detective gathering evidence, helps ensure your understanding of the concept. By checking derivations and simplifications, you're extending your skills to critically evaluate functions beyond mere computation.