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Chapter 1: Problem 46

Identify any intercepts and test for symmetry. Then sketch the graph of the equation. $$y=2 x-3$$

Short Answer

Expert verified
The graph of the equation \(y = 2x-3\) has a y-intercept at -3 and x-intercept at 1.5. There is no symmetry. The graph is a straight line passing through points (1.5, 0) and (0, -3).

Step by step solution

01

Identify the Y-intercept

The y-intercept is the point on the graph where the line crosses the y-axis. This occurs when x=0. For the equation \(y=2x-3\), when x=0, \(y = -3\). So, the y-intercept is -3.
02

Identify the X-intercept

The x-intercept is the point on the graph where it crosses the x-axis. This happens when y=0. Set y=0 in the equation \(0=2x-3\), on solving we find that x = 3/2 or 1.5. This means the x-intercept is 1.5.
03

Test for Symmetry

Considering symmetry in the y-axis, when we replace \(x\) with \(-x\), if the equation remains the same then it shows symmetry. On replacement in the given equation, we get \(y = 2(-x) -3 = -2x - 3\), which is not the same as the original equation. So, the equation is not symmetric about the y-axis. Likewise, for symmetry in the origin, replace both \(x\) and \(y\) with \(-x\) and \(-y\) respectively. The equation should remain the same as the original equation, but in this case, -y = 2(-x) -3 turns into \(y = 2x + 3\), which again is not the same. Hence, the equation has no symmetry over the y-axis or the origin.
04

Sketch the Graph

Using the x and y intercepts plot (1.5, 0) and (0, -3) on the graph. Draw a line passing through these points which extends infinitely in both directions. The straight line represents the equation \(y = 2x - 3\).

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Key Concepts

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

Intercepts
In the context of a linear equation such as \(y=2x-3\), the intercepts are essential points where the graph of the line crosses the axes of the coordinate plane. Let's explore these intercepts:

  • Y-intercept: This is the point where the line crosses the y-axis. To find the y-intercept, we set \(x=0\) in the equation. For our equation, substituting \(x = 0\) gives \(y = 2(0) - 3 = -3\). Therefore, the y-intercept is the point \((0, -3)\).
  • X-intercept: This point is located where the line crosses the x-axis. To find the x-intercept, we set \(y=0\) in the equation. Substituting gives \(0 = 2x - 3\), solving this gives \(x = \frac{3}{2} = 1.5\). Therefore, the x-intercept is the point \((1.5, 0)\).
Intercepts are fundamental because they give us key reference points for drawing the line on a graph. Knowing just these two points allows us to visualize and sketch the entire line effectively.
Linearity
The term "linearity" indicates that the relationship between the variables in the equation is consistent and direct, forming a straight line when graphed. This is an important feature of linear equations such as \(y = 2x - 3\). Here's why linearity is significant:

  • Linear equations represent constant rates of change. For every unit increase in \(x\), there is a consistent increase or decrease in \(y\). In our equation, for every 1 unit increase in \(x\), \(y\) increases by 2 units, as shown by the coefficient of \(x\).
  • The graph of a linear equation is always a straight line. This characteristic simplifies the process of graphing because only two points are needed to determine the entire line.
The equation \(y = 2x - 3\) provides a straightforward example of linearity, with its slope of 2 and y-intercept of -3, giving a clear and predictable graphical representation.
Graph Symmetry
Graph symmetry is about whether a graph is mirrored around an axis or a point. For the equation \(y=2x-3\), checking for symmetry helps us understand the graph's behavior:

  • Y-axis Symmetry: A graph has y-axis symmetry if replacing \(x\) with \(-x\) in its equation results in the original equation. Substituting \(-x\) into our equation gives \(y = -2x - 3\). This is not the same as the original equation, so the graph is not symmetric around the y-axis.
  • Origin Symmetry: A graph has origin symmetry if replacing both \(x\) with \(-x\) and \(y\) with \(-y\) leaves the equation unchanged. However, when we substitute in our equation, the result is \(-y = -2x - 3\), or equivalently \(y = 2x + 3\), which does not match the original. Thus, there is no symmetry about the origin either.
Symmetry analyses can reveal much about graph properties, but here it highlights the asymmetrical nature of \(y = 2x - 3\), indicating the uniqueness of its trajectory on the plane without mirroring effects.

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Most popular questions from this chapter

Decide whether the statement is true or false. Justify your answer. In the equation for the area of a circle, \(A=\pi r^{2},\) the area \(A\) varies jointly with \(\pi\) and the square of the radius \(r\).

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Determine whether the statement is true or false. Justify your answer. When you are given two functions \(f(x)\) and \(g(x),\) you can calculate \((f \circ g)(x)\) if and only if the range of \(g\) is a subset of the domain of \(f.\)

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\sqrt{2 x+3}$$
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