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Chapter 2: Problem 59

Sketch the graph of the equation. Identify any intercepts and test for symmetry. \(y=5-3 x\)

Short Answer

Expert verified
The graph of the equation \(y=5-3x\) is a straight line with a negative slope. It crosses the y-axis at (0,5) and the x-axis at approximately (\(\frac{5}{3}\),0). The graph doesn't show any symmetry.

Step by step solution

01

Sketch the Graph

Start by identifying the y-intercept which is the constant term, 5. This is the point where the line crosses the y-axis. So we have the point (0,5). The slope of the line (which is -3) tells us that for each positive step along the x-axis, we go three steps negatively along the y-axis. Draw a straight line through these points which will represent the equation \(y=5-3x\).
02

Identify the Intercepts

The y-intercept is already found at point (0,5). To find the x-intercept(s), set y equal to 0 and solve for x. So, \(0=5-3x\) gives \(x=\frac{5}{3}\) or approximately 1.67. So, the x-intercept is at the point (\(\frac{5}{3}\),0).
03

Test for Symmetry

A function is symmetric about the y-axis if replacing \(x\) with \(-x\) results in the same function, and it's symmetric about the origin if that replacement yields the negative of the original function. Replace \(x\) with \(-x\) in the equation. The equation becomes \(y=5-(-3x) = 5+3x\), which is not the same as the original function. So, the graph has no symmetry about the y-axis or the origin.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Sketching
When sketching the graph of a linear equation, such as \( y = 5 - 3x \), we begin by identifying crucial points like intercepts, and then drawing the line defined by the equation. Linear equations result in straight lines. Thus, having a few key points is usually enough to define the graph.
  • Identify the Intercepts: We'll start by finding the y-intercept and x-intercepts (if any). These points help us accurately plot the line on a graph.
  • Understanding the Slope: The equation \( y = 5 - 3x \) reveals a slope of \(-3\). This means as you move 1 unit right (positive direction on the x-axis), the line will drop 3 units down (negative direction on the y-axis), creating a line that slants downward from left to right.
  • Plotting and Drawing: Begin plotting from the y-intercept at (0,5). Move according to the slope until at least two points are set. Connect these points with a straight line to represent the equation.
A sketch provides visual insight into the behavior of the equation over different x-values.
Intercepts
Intercepts are critical points where the graph intersects the axes. Specifically, we'll look at:
  • Y-intercept: This is where the line crosses the y-axis. In the equation \( y = 5 - 3x \), the constant term, \(5\), indicates that the y-intercept is at the point (0,5).
  • X-intercept: Found by setting \(y = 0\) and solving for \(x\). Substituting into our equation, we get \(0 = 5 - 3x\). Solving for \(x\), we find the x-intercept at \(x = \frac{5}{3}\) or approximately 1.67, making the x-intercept at (\(\frac{5}{3}\), 0).
Identifying these points helps in drawing an accurate graph. They show the exact spots where the line will meet each axis.
Slope
The slope of a linear equation like \( y = 5 - 3x \) is a measure of how steep the line is. In simple terms, it tells us how much \( y \) increases or decreases as \( x \) increases by one unit.
  • Understanding Slope: With the equation in the form \( y = mx + b \), \(m\) is the slope. Here, \(m = -3\).
  • Interpreting \(-3\): This slope means for every unit step to the right on the x-axis, the line drops 3 steps. It is negative, indicating a downward slant from left to right.
  • Practical Use: Start from the y-intercept (0,5), then use the slope: move 1 unit to the right, and 3 units down to locate the next point.
Calculating and understanding slope if fundamental for predicting line behavior and plotting accordingly.
Symmetry
Symmetry in the context of graphing refers to whether a graph can be reflected across an axis or around a point without changing appearance.
  • Y-axis Symmetry: A graph has symmetry about the y-axis if replacing \(x\) with \(-x\) yields the same equation. For \( y = 5 - 3x \), replacing \(x\) with \(-x\) results in \( y = 5 + 3x \), which changes the equation, indicating no y-axis symmetry.
  • Origin Symmetry: Symmetry about the origin requires that replacing \(x\) with \(-x\) and \(y\) with \(-y\) results in the same equation. Using our equation, it becomes \(-y = 5 + 3x\) after necessary changes, which isn't equivalent to the original.
Therefore, the graph of \( y = 5 - 3x \) exhibits neither y-axis nor origin symmetry, affecting the predictability of the graph's shape.

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Most popular questions from this chapter

The average annual price-earnings ratio for a corporation's stock is defined as the average price of the stock divided by the earnings per share. The average price of a corporation's stock is given as the function \(P\) and the earnings per share is given as the function \(E\). Find the price-earnings ratios, \(P / E\), for the years 2001 to 2005 . Jack in the Box $$\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 2001 & 2002 & 2003 & 2004 & 2005 \\\\\hline P & \$ 27.22 & \$ 28.19 & \$ 19.38 & \$ 25.20 & \$ 36.21 \\\\\hline E & \$ 2.11 & \$ 2.33 & \$ 2.04 & \$ 2.27 & \$ 2.48 \\\\\hline\end{array}$$

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