Problem 14 - Algebra and Trigonometry

Question

\(11-36\) Make a table of values and sketch the graph of the equation. Find the \(x\) - and \(y\) -intercepts. $$ y=3 x+3 $$

Step-by-Step Solution

Answer

x-intercept: (-1, 0), y-intercept: (0, 3).

Step 1: Choose values for x

To create a table of values, select several values for \(x\). It's common to choose values like \(-2, -1, 0, 1,\) and \(2\) to generate points for the graph.

Step 2: Calculate corresponding y values

Use the equation \(y = 3x + 3\) to calculate \(y\) for each chosen \(x\). For example:- If \(x = -2\), then \(y = 3(-2) + 3 = -6 + 3 = -3\)- If \(x = -1\), then \(y = 3(-1) + 3 = -3 + 3 = 0\)- If \(x = 0\), then \(y = 3(0) + 3 = 3\)- If \(x = 1\), then \(y = 3(1) + 3 = 3 + 3 = 6\)- If \(x = 2\), then \(y = 3(2) + 3 = 6 + 3 = 9\).

Step 3: Create the table of values

Organize the \(x\) and \(y\) values into a table:\[\begin{array}{c|c} x & y \\hline-2 & -3 \-1 & 0 \0 & 3 \1 & 6 \2 & 9 \\end{array}\]

Step 4: Plot the points

Using the table of values, plot each point \((-2, -3), (-1, 0), (0, 3), (1, 6), (2, 9)\) on a coordinate graph. Ensure each point is accurately placed according to its \(x\) and \(y\) values.

Step 5: Draw the graph

Once the points are plotted, draw a straight line through them, as the equation \(y = 3x + 3\) is linear. Extend the line across the graph to represent all possible solutions.

Step 6: Find x-intercept

The \(x\)-intercept occurs where \(y = 0\). Set \(y = 0\) in the equation \(y = 3x + 3\): \[0 = 3x + 3\] Subtract 3 from both sides:\[-3 = 3x\]Divide by 3:\[x = -1\]So, the \(x\)-intercept is \((-1, 0)\).

Step 7: Find y-intercept

The \(y\)-intercept occurs where \(x = 0\). Substitute \(x = 0\) into the equation \(y = 3x + 3\): \[y = 3(0) + 3 = 3\]So, the \(y\)-intercept is \((0, 3)\).

Key Concepts

Graphing Linear Functions

Graphing linear functions is a fundamental skill in algebra that involves plotting points on a graph to form a straight line. The equation given, \(y = 3x + 3\), is in the slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This equation tells us that the line has a slope of 3, meaning it rises three units for every one unit it moves to the right. Additionally, the y-intercept is 3, indicating that the line crosses the y-axis at \(y = 3\).

To graph a linear equation like this one, you simply need the slope and one point, typically the y-intercept. However, ensuring accuracy can be easier with a table of values and plotting multiple points. Once the points are plotted, draw a straight line connecting them. This line represents all solutions to the equation.

x-intercept

The x-intercept is a specific point on a graph where the line crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept of the equation \(y = 3x + 3\), you substitute \(y = 0\) into the equation and solve for \(x\).

In this case:

• Set \(y = 0\) in the equation: \(0 = 3x + 3\).
• Subtract 3 from both sides: \(-3 = 3x\).
• Divide by 3: \(x = -1\).

Thus, the x-intercept is at the point \((-1, 0)\). This means that the graph will cross the x-axis at \(x = -1\). Recognizing the x-intercept provides valuable insight into where the line will interact with the x-axis.

y-intercept

The y-intercept is the point where the graph crosses the y-axis, and at this point, the value of \(x\) is zero. To find the y-intercept, substitute \(x = 0\) into the equation. For the given linear equation \(y = 3x + 3\), the process is straightforward:

• Substitute \(x = 0\) into the equation: \(y = 3(0) + 3\).
• Simplify to find \(y = 3\).

Therefore, the y-intercept is at the point \((0, 3)\). This tells us where the line meets the y-axis, which is a crucial part of graphing the function accurately. Knowing the y-intercept helps in drawing the line since it provides a starting point on the graph.

Table of Values

Creating a table of values is a practical method to plot several points of a linear function, ensuring precision in graphing. This technique involves selecting a range of \(x\) values and using the equation to find corresponding \(y\) values. For example, using \(y = 3x + 3\), you calculate:

• When \(x = -2\), then \(y = 3(-2) + 3 = -3\).
• When \(x = -1\), then \(y = 3(-1) + 3 = 0\).
• When \(x = 0\), then \(y = 3(0) + 3 = 3\).
• When \(x = 1\), then \(y = 3(1) + 3 = 6\).
• When \(x = 2\), then \(y = 3(2) + 3 = 9\).

Once you have these points \((-2, -3), (-1, 0), (0, 3), (1, 6), (2, 9)\), you can plot them on a coordinate graph. Drawing a line through these plotted points yields the linear function's graph, demonstrating how easily a table of values can facilitate accurate graphing of linear equations.