Problem 103 - College Algebra

Question

Graph the linear function \(f\) on a domain of [-0.1,0.1] for the function whose slope is 75 and \(y\) -intercept is -22.5. Label the points for the input values of -0.1 and 0.1 .

Step-by-Step Solution

Answer

Plot the points (-0.1, -30) and (0.1, -15), and draw a line through them.

Step 1: Write the Equation of the Line

We know the general form of a linear function is given by \(f(x) = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. Given that the slope \(m=75\) and the \(y\)-intercept \(c=-22.5\), the equation of the line is \(f(x) = 75x - 22.5\).

Step 2: Calculate the Value of the Function at x = -0.1

Substitute \(x = -0.1\) into the function \(f(x) = 75x - 22.5\): \[f(-0.1) = 75(-0.1) - 22.5 = -7.5 - 22.5 = -30\]So the coordinate for \(x = -0.1\) is \((-0.1, -30)\).

Step 3: Calculate the Value of the Function at x = 0.1

Substitute \(x = 0.1\) into the function \(f(x) = 75x - 22.5\):\[f(0.1) = 75(0.1) - 22.5 = 7.5 - 22.5 = -15\]So the coordinate for \(x = 0.1\) is \((0.1, -15)\).

Step 4: Determine the Points for the Graph

We have two key points for the graph: \((-0.1, -30)\) and \((0.1, -15)\). These points will be used to draw the line over the domain \([-0.1, 0.1]\).

Step 5: Graph the Function

To graph the function, plot the points \((-0.1, -30)\) and \((0.1, -15)\) on a coordinate plane. Draw a straight line passing through these two points, ensuring the line does not extend beyond the domain \([-0.1, 0.1]\). Remember to label these points for clarity.

Key Concepts

Graphing

Graphing a linear function is a visual representation of its equation. To create a graph, you begin by identifying points that the line will pass through. Here's how you do it:

• Identify the equation of the function, in this case, that is given by the formula: \(f(x) = 75x - 22.5\).
• Determine key points by substituting values into the equation. For example, for \(x = -0.1\), the corresponding \(y\) value is \(-30\), giving the coordinate \((-0.1, -30)\). For \(x = 0.1\), the coordinate is \((0.1, -15)\).
• Plot these coordinates on a chart, marking them clearly to improve understanding.
• Connect the points with a straight line, which is representative of the equation throughout the given domain.

Graphing helps in understanding the behavior or trend depicted by the linear function. This visual tool provides a straightforward path to engaging with data.

Slope

The slope of a line indicates its steepness and direction. In the function \(f(x) = 75x - 22.5\), the slope is \(75\). Here's why the slope is a key concept:

• A positive slope like \(75\) shows that the line ascends as you move from left to right. This implies that for every unit increase in \(x\), the \(y\) value increases by \(75\).
• The steeper the slope, the more vertical the line appears. A slope of \(75\) means quite a steep line due to the large amount increase in \(y\) for minimal changes in \(x\).
• In mathematical terms, the slope \(m\) is calculated as \(m = (f(x_2) - f(x_1)) / (x_2 - x_1)\), which reflects the change in \(y\) over the change in \(x\).

Understanding slope will enhance your prediction of how the graph behaves when alterations in \(x\) occur.

Y-Intercept

The \(y\)-intercept is the point where a line crosses the \(y\)-axis. For the equation \(f(x) = 75x - 22.5\), the \(y\)-intercept is \(-22.5\). Here’s why this is significant:

• It represents the starting point value of \(y\) when \(x = 0\). In other words, it shows what the value of the output is when no other input has been added.
• A negative \(y\)-intercept like \(-22.5\) means the line started below the \(x\)-axis and moves upwards due to the positive slope.
• Locating the \(y\)-intercept on the graph means finding the point directly on the \(y\)-axis, showing where and how a line begins its ascent or descent.

Understanding the \(y\)-intercept aids in forming a complete picture of the linear equation's trajectory.

Coordinates

Coordinates are a set of values that show the exact position of a point on a graph. For our example, coordinates include points like \((-0.1, -30)\) and \((0.1, -15)\). Here’s how coordinates simplify graph interpretation:

• Each coordinate is written as \((x, y)\), where \(x\) denotes the horizontal position, and \(y\) indicates the vertical position on the graph.
• To graph a linear equation, coordinates are critical as they exemplify exactly where the line should cross through the plane.
• Knowing how to identify and plot these points is fundamental for visually representing mathematical relations on a graph.

Using coordinates allows for an easier recognition of patterns, making the graph a helpful guide in data interpretation.