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Chapter 2: Problem 90

For the following exercises, find the x- and y-intercepts of each equation $$h(x)=3 x-5$$

Short Answer

Expert verified
X-intercept: \( \left( \frac{5}{3}, 0 \right) \), Y-intercept: \( (0, -5) \).

Step by step solution

01

Identify the Equation

The given function is \( h(x) = 3x - 5 \). This is a linear equation, where the goal is to find where this line crosses the axes, meaning we need to determine both the x-intercept and the y-intercept.
02

Find the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation. This gives \( h(0) = 3(0) - 5 = -5 \). Hence, the y-intercept is at the point \((0, -5)\).
03

Find the X-Intercept

To find the x-intercept, set \( h(x) = 0 \). Thus, solve \( 3x - 5 = 0 \) for \( x \). Add 5 to both sides to get \( 3x = 5 \), and then divide by 3 to get \( x = \frac{5}{3} \). Therefore, the x-intercept is at the point \( \left( \frac{5}{3}, 0 \right) \).

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Key Concepts

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

x-intercept
In the context of a linear equation, finding the x-intercept means identifying the point where the graph of the equation crosses the x-axis. To determine this point, you set the output of the function to zero because the y-coordinate of any point on the x-axis is always zero. For the equation in the exercise, which is \( h(x) = 3x - 5 \), you set \( h(x) = 0 \).

Here's how to find the x-intercept step-by-step:
  • Set the equation equal to zero: \( 3x - 5 = 0 \).
  • Solve for \( x \) by isolating the variable: First add 5 to both sides to get \( 3x = 5 \). Then, divide by 3 to find \( x = \frac{5}{3} \).
The x-intercept is at the point \( \left( \frac{5}{3}, 0 \right) \). This means that at \( x = \frac{5}{3} \), the function \( h(x) \) crosses the x-axis.
y-intercept
The y-intercept of a linear equation is the point where the graph crosses the y-axis. To find this point, you set the input \( x = 0 \) and solve for \( h(x) \), which gives you the y-coordinate of the intercept. The reason we set \( x = 0 \) is because every point on the y-axis has an \( x \)-value of zero. Let鈥檚 see how it鈥檚 done with the equation \( h(x) = 3x - 5 \):

  • Substitute \( x = 0 \) into the equation: \( h(0) = 3(0) - 5 \).
  • Simplify the expression to find \( h(0) = -5 \).
This calculation shows that the y-intercept is at the point \( (0, -5) \). This means that when \( x = 0 \), the value of the function \( h(x) \) is \(-5\), and the graph crosses the y-axis at this point.
function analysis
Function analysis involves studying various aspects of a function, including its intercepts, slope, and overall behavior. For a linear function such as \( h(x) = 3x - 5 \), this process is straightforward yet insightful.

When analyzing a linear function:
  • The slope, derived from the coefficient of \( x \), indicates the rate of change. Here, the slope is \(3\), meaning for every unit increase in \( x \), \( h(x) \) increases by \(3\) units.
  • The y-intercept is the point where the line crosses the y-axis, marking where the graph starts when \( x = 0 \). In this case, it's \(-5\).
  • The x-intercept is where the graph touches the x-axis, given by \( \left( \frac{5}{3}, 0 \right) \).
Understanding these attributes aids in sketching the graph, predicting values, and interpreting the linear relationship's implication in real-world scenarios. Recognizing these aspects allows for a comprehensive analysis of the function.

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Most popular questions from this chapter

When hired at a new job selling electronics, you are given two pay options: \(\cdot\) Option A: Base salary of \(\$ 10,000\) a year with a commission of 4\(\%\) of your sales \(\cdot\) Option B: Base salary of \(\$ 20,000\) a year with a commission of 4\(\%\) of your sales How much electronics would you need to sell for option A to produce a larger income?

A town鈥檚 population increases at a constant rate. In 2010 the population was 55,000. By 2012 the population had increased to 76,000. If this trend continues, predict the population in 2016.

The number of people afflicted with the common cold in the winter months dropped steadily by 50 each year since 2004 until 2010. In 2004, 875 people were inflicted. Find the linear function that models the number of people afficted with the common cold \(C\) as a function of the year, \(t\) When will no one be befficted?

Graph the linear function \(f\) on a domain of \([-0.1,0.1]\) for the function whose slope is 75 and \(y\) -intercept is \(-22.5 .\) Label the points for the input values of \(-0.1\) and \(0.1 .\)

Eight students were asked to estimate their score on a 10-point quiz. Their estimated and actual scores are given in Table 2.17. Plot the points, then sketch a line that fits the data. $$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline \text { Predicted } & {6} & {7} & {7} & {8} & {7} & {9} & {10} & {10} \\ \hline \text { Actual } & {6} & {7} & {8} & {8} & {9} & {10} & {10} & {9} \\ \hline\end{array}$$
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