Question

If \(f(x)=-3 x+10,\) then the graph of \(f\) is a________ with slope _________and y-intercept _________.

Short answer

The graph is a straight line with slope \(-3\) and y-intercept \(10\).

Step-by-step solution

- Identify the Function Form

The function given is in the form of a linear equation: \(f(x) = -3x + 10\). The standard form of a linear equation is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

- Determine the Type of Graph

Since the function is a linear equation, the graph of \(f(x) = -3x + 10\) is a straight line.

- Find the Slope

Comparing the given function \(f(x) = -3x + 10\) to the standard form \(y = mx + b\), it is clear that the coefficient of \(x\) is \(-3\). Hence, the slope \(m\) is \(-3\).

- Find the Y-Intercept

The constant term in the linear function \(f(x) = -3x + 10\) is \(10\). Hence, the y-intercept \(b\) is \(10\).

- Write the Final Answer

The graph of \(f(x) = -3x + 10\) is a straight line with a slope of \(-3\) and a y-intercept of \(10\).

Key concepts

Slope

Understanding the slope of a line is essential in graphing linear functions. The slope, often denoted as \(m\) in the equation \(y = mx + b\), measures the steepness of the line. In simpler terms, it tells you how much y changes for a unit change in x. For the given function \(f(x) = -3x + 10\), the slope is \(-3\).

A negative slope means the line slopes downwards as it moves from left to right. Here, the line falls by 3 units for every 1 unit it moves to the right. Think of it as skiing down a steep hill; the greater the negative value, the steeper the hill.

To summarize, the slope:

• Indicates whether the line rises or falls
• Is calculated as the change in y over the change in x (rise over run)
• For \(f(x) = -3x + 10\), the slope is \(-3\)

Y-Intercept

The y-intercept is another crucial element of a linear equation. This is the point where the line crosses the y-axis. In the equation \(y = mx + b\), the y-intercept is represented by the constant term \(b\). For the equation \(f(x) = -3x + 10\), the y-intercept is 10.

This means when \(x = 0\), \(f(x) = 10\). You can easily spot this point on the graph of the function: it's where the line meets the y-axis.

Key points about the y-intercept:

• It occurs where \(x = 0\)
• For \(f(x) = -3x + 10\), the y-intercept is \(10\)
• This helps to quickly locate a starting point for graphing the line

Graphing Linear Functions

Graphing linear functions can seem intimidating, but it becomes straightforward when you understand the concepts of slope and y-intercept. The equation \(f(x) = -3x + 10\) provides all the information you need.

Start by plotting the y-intercept on the graph, which is at point \((0, 10)\). From this point, use the slope to determine the next points. Since the slope is \(-3\), move 1 unit to the right (positive x-direction) and 3 units down (negative y-direction). You can repeat this step to plot multiple points if needed.

Connect these points with a straight line, and you'll have the graph of your linear function.

Key steps for graphing linear functions:

• Identify the y-intercept and plot it
• Use the slope to find other points
• Draw a straight line through the points

Following these steps will make graphing linear functions much more manageable.