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Chapter 2: Problem 39

Graph each equation by hand. $$y=5-10 x, y=|5-10 x|$$

Short Answer

Expert verified
Draw a straight line for \( y = 5-10x \). For \( y = |5-10x| \), make a V-shape using the point (0.5, 0) as the vertex.

Step by step solution

01

Identify Forms of Given Equations

The first equation is a linear equation of the form \( y = mx + c \), where \( m = -10 \) and \( c = 5 \). The second equation involves an absolute value function, which modifies the linear equation into an absolute value equation \( y = |5 - 10x| \).
02

Graph the Linear Equation

Graph the equation \( y = 5 - 10x \). This is a straight line with a y-intercept at (0,5) and a slope of -10, meaning it drops 10 units for every 1 unit it moves to the right.
03

Plot the Line's Important Points

Find two points from the linear equation to draw the line accurately. Use the y-intercept (0, 5). Find another point by setting \( y = 0 \) and solve for \( x \): \( 5 - 10x = 0 \) gives \( x = 0.5 \). So, another point is (0.5, 0).
04

Draw the Absolute Value Graph

The graph of \( y = |5 - 10x| \) is derived from the line \( y = 5 - 10x \) by reflecting the negative portion of the line over the x-axis. Plot the vertex of the V-shape, which occurs where \( 5 - 10x = 0 \), at \( (0.5, 0) \).
05

Apply the V-Shape of Absolute Functions

Draw the V-shape by using the vertex at \( (0.5, 0) \). The portion of the line where \( y = 5 - 10x \) is positive remains unchanged, while the negative values are mirrored across the x-axis. Extend the lines from the vertex in both directions with the same slopes, \( -10 \) to the left and \( 10 \) to the right of \( x = 0.5 \).
06

Verify the Graph

Check key points on both graphs. For \( x = 0 \), \( y = 5 \) for both equations. For \( x = 1 \), \( y = -5 \) for \( y = 5-10x \) and \( y = 5 \) for \( |5 - 10x| \). These points should match the graph.

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Key Concepts

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

Linear Equation
A linear equation forms the backbone of graphing linear functions. It can be represented in the form of \( y = mx + c \), where \( m \) is the slope and \( c \) the y-intercept. Linear equations create straight lines when graphed on a coordinate plane.
The equation \( y = 5 - 10x \) is an example. It is a simple line that shows the relationship between \( x \) and \( y \).
Due to its predictable nature, it is widely used in elementary algebra.
  • Each point \( (x, y) \) on the line satisfies the equation.
  • The line continues infinitely in both directions unless restricted by a specific domain.
  • The y-intercept is the point where the line crosses the y-axis.
  • The slope decides the steepness and direction of the line.
Absolute Value
Absolute value equations add a twist to linear relationships by introducing the concept of magnitude, which makes every output positive. In essence, an absolute value, denoted by vertical bars as in \( |x| \), measures how far a number is from zero without considering its direction.
It has profound effects on graphing since negative parts "flip" upwards, creating a distinctive V-shape.

Here's how it changes equations:
  • In the absolute value equation \( y = |5 - 10x| \), sections beneath the x-axis reflect upwards.
  • This reflection creates two mirror images of the portion of the line above the x-axis.
  • The vertex at \( (0.5, 0) \) is the point where the graph switches from negative to positive.
  • The graph now appears as two intersecting lines stemming from the vertex.
Understanding this helps recognize graphs involving absolute functions immediately.
Slope
The slope is a crucial concept in graphing linear equations as it determines how a line spans across the coordinate plane. Defined as "rise over run," it compares vertical movement to horizontal movement. In the equation \( y = 5 - 10x \), the slope is \(-10\). This means the line falls steeply downward. For any 1 unit increase in \(x\), \(y\) decreases by 10 units.
  • A positive slope slopes upwards as \(x\) increases.
  • A negative slope, like in this case, slants downwards, creating a descending line.
  • A greater numerical value in the slope means a steeper line.
  • If the slope is zero, the line is horizontal.
Recognizing the impact of the slope makes graph prediction much easier. It serves as a visual cue to the direction and rate of change on a graph.
Y-intercept
The y-intercept serves as the point where a line crosses the y-axis, providing information about a graph's starting point vertically. For the linear equation \( y = 5 - 10x \), the y-intercept is \(5\). This is the value of \(y\) when \(x\) is zero.
  • The y-intercept provides a starting point for graphing a line.
  • It is marked by the coordinates \((0, c)\), where \(c\) is the intercept value.
  • On a graph, it often indicates an initial condition or beginning state.
  • The y-intercept can be quickly found by inspecting the equation's constant term.
Understanding this helps initiate sketching a line graph and builds a complete understanding of how the equation behaves.

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Most popular questions from this chapter

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=x^{2}+2 x$$

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=4 x+3$$

The greenhouse gas methane lets sunlight into the atmosphere, but blocks heat from escaping the earth's atmosphere. Methane is a by-product of buming fossil fuels. In the table, \(f\) models the predicted methane emissions in millions of tons produced by developed countries during year \(x\). The function \(g\) models the same emissions for developing countries. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 1990 & 2000 & 2010 & 2020 & 2030 \\\\\hline f(x) & 27 & 28 & 29 & 30 & 31 \\\\\hline g(x) & 5 & 7.5 & 10 & 12.5 & 15\end{array}$$ (a) Make a table for a function \(h\) that models the total predicted methane emissions for developed and developing countries. (b) Write an equation that relates \(f(x), g(x),\) and \(h(x)\)

Use the analyric method of Example 3 to determine whether the graph of the given function is symmetric with respect to the \(y\) -axis, symmetric with respect to the origin, or neither. Use \(a\) calculator and the standand window to support your conclusion. $$f(x)=x^{5}-2 x^{3}$$

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-10| \leq 36,\) Chesterfield, Canada
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