Problem 22 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=3 x+3 $$

Step-by-Step Solution

Answer

The x-intercept is (-1, 0) and the y-intercept is (0, 3). The graph has no symmetry.

Step 1: Identify the Equation

The given equation is a linear equation: \( y = 3x + 3 \). This equation represents a straight line.

Step 2: Make a Table of Values

To sketch the graph, let's create a table of values by choosing some values for \( x \) and calculating the corresponding \( y \) values. | \( x \) | \( y = 3x + 3 \) ||---|---|| -2 | \( y = 3(-2) + 3 = -6 + 3 = -3 \) || -1 | \( y = 3(-1) + 3 = -3 + 3 = 0 \) || 0 | \( y = 3(0) + 3 = 0 + 3 = 3 \) || 1 | \( y = 3(1) + 3 = 3 + 3 = 6 \) || 2 | \( y = 3(2) + 3 = 6 + 3 = 9 \) |

Step 3: Find the x-Intercept

The x-intercept occurs when \( y = 0 \). Set the equation to zero and solve for \( x \). \[ 0 = 3x + 3 \] \[ 3x = -3 \] \[ x = -1 \] So, the x-intercept is \( (-1, 0) \).

Step 4: Find the y-Intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) in the equation. \( y = 3(0) + 3 = 3 \). So, the y-intercept is \( (0, 3) \).

Step 5: Check for Line Symmetry

For checking symmetry, we have three types to consider:- **Symmetry about the y-axis**: Replace \( x \) with \( -x \). The equation becomes \( y = 3(-x) + 3 = -3x + 3 \), which is not equal to \( y = 3x + 3 \), hence it is not symmetric about the y-axis.- **Symmetry about the x-axis**: Replace \( y \) with \( -y \). The equation becomes \( -y = 3x + 3 \), which is not equal to \( y = 3x + 3 \), hence it is not symmetric about the x-axis.- **Symmetry about the origin**: Replace \( x \) with \( -x \) and \( y \) with \( -y \). The equation becomes \( -y = 3(-x) + 3 = -3x + 3 \), which again is not equal to \( y = 3x + 3 \), hence it is not symmetric about the origin.

Step 6: Sketch the Graph

Using the table of values and the intercepts, plot the points (\(-2, -3\)), (\(-1, 0\)), (0, 3), (1, 6), and (2, 9) on a graph. Draw a straight line through these points. This line makes a graph of the equation \( y = 3x + 3 \).

Key Concepts

Graphing Linear Equations

When we talk about graphing linear equations, we're diving into plotting points that form a straight line on a graph. For instance, the equation \( y = 3x + 3 \) represents a classic linear equation. The key to graphing involves a few simple steps:

• Create a table of values: Select specific values of \( x \) and use the equation to find \( y \).
• Plot these (\( x, y \)) points on a coordinate plane.
• Draw a line through all the plotted points to complete the graph.

This approach helps visualize how the values of \( x \) and \( y \) are connected. It's like connecting dots where each dot is one particular value pair from your table. For \( y = 3x + 3 \), the line will slope upwards because the coefficient of \( x \) (which is 3) is positive.

x-Intercept

Finding the x-intercept of a linear equation gives insight into where the line crosses the x-axis. This is a point where the value of \( y \) is zero. To determine this intercept for any equation of the form \( y = mx + b \), you set \( y = 0 \) and solve for \( x \).

For the equation \( y = 3x + 3 \):
When \( y = 0 \), it becomes \( 0 = 3x + 3 \). Solving this:

• Subtract 3 from both sides to get \( 3x = -3 \).
• Divide both sides by 3 to find \( x = -1 \).

Thus, the x-intercept is the point \((-1, 0)\). It tells us that the line crosses the x-axis at \( x = -1 \). This point is crucial for accurately drawing the line, as it signals how the graph interacts with one of the axes.

y-Intercept

The y-intercept is an essential part of graphing a linear equation, as it shows where the line crosses the y-axis. For any linear equation \( y = mx + b \), the y-intercept is simply the constant term \( b \), representing the point where \( x \) is zero.

For \( y = 3x + 3 \):

• When \( x = 0 \), substitute into the equation: \( y = 3(0) + 3 = 3 \).

This gives us a y-intercept at the point \((0, 3)\). It indicates that the line crosses the y-axis above the origin at \( y = 3 \). This intercept is crucial, especially when starting to sketch the graph, as it provides one fixed point to draw the line through.

Symmetry in Graphs

Symmetry in graphs reveals if and how a graph reflects across certain lines or points. For linear equations like \( y = 3x + 3 \), we often check three types of symmetry:

• **Symmetry about the y-axis**: If replacing \( x \) with \( -x \) leaves the equation unchanged, it's symmetric about the y-axis. For \( y = 3x + 3 \), replacing \( x \) results in \( y = -3x + 3 \), which is different. Thus, no y-axis symmetry.
• **Symmetry about the x-axis**: Replace \( y \) with \( -y \). If the equation remains unchanged, it is symmetric about the x-axis. Here, \( y = 3x + 3 \) becomes \( -y = 3x + 3 \), which does not lead to equality. Therefore, there is no x-axis symmetry.
• **Symmetry about the origin**: Check by replacing both \( x \) with \( -x \) and \( y \) with \( -y \). For our equation, this substitution results in \( -y = -3x + 3 \), violating symmetry conditions about the origin.

Understanding these symmetries helps determine how a graph extends and mirrors across axes and the origin.