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Chapter 5: Problem 1

What are the intercepts of \(y=5 x+10 ?\)

Short Answer

Expert verified
The intercepts are (0, 10) and (-2, 0).

Step by step solution

01

Identify the y-intercept

To find the y-intercept, set the value of x to 0 in the equation. Solve for y.
02

Calculate the y-intercept

Substitute x = 0 into the equation: \[ y = 5(0) + 10 \] This simplifies to \[ y = 10 \] Thus, the y-intercept is (0, 10).
03

Identify the x-intercept

To find the x-intercept, set the value of y to 0 in the equation. Solve for x.
04

Calculate the x-intercept

Substitute y = 0 into the equation: \[ 0 = 5x + 10 \] Solve for x: \[ -10 = 5x \] \[ x = -2 \] Thus, the x-intercept is (-2, 0).

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Key Concepts

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

y-intercept
The y-intercept is the point where a line or curve crosses the y-axis on a graph. To find the y-intercept of a linear equation, you need to set the value of x to 0 and solve for y. When x is zero, any term involving x will disappear, simplifying your calculations. For example, in the equation provided, you substitute x with 0:
  1. Substitute x = 0: 
     y = 5(0) + 10 = 10
  2. Therefore, the y-intercept is (0, 10) because when x is 0, y is 10.
Understanding intercepts helps in graphing and interpreting functions. It's worth noting that the y-intercept signifies the starting position of a line when graphed.
x-intercept
The x-intercept is the point where a line or curve crosses the x-axis on a graph. To find the x-intercept of a linear equation, set the value of y to 0 and solve for x. When y is zero, you eliminate the y term, which simplifies the equation. In the provided example, you find the x-intercept as follows:
  1. Substitute y = 0: 
     0 = 5x + 10
  2. Solve for x: 
     -10 = 5x  x = -2
This indicates that the x-intercept is (-2, 0) because when y is 0, x is -2. Skipping this step might result in errors in graphing or interpreting functions.
linear equation
A linear equation is an equation that makes a straight line when it is graphed. These equations can be written in the standard form y = mx + b, where m is the slope and b is the y-intercept. Here are some essential points to keep in mind:
  • Linear equations describe a constant rate of change.
  • They are characterized by their straight-line graphs.
  • The slope (m) represents the steepness of the line and direction, either positive or negative.
In the given example, the equation y = 5x + 10 has a slope of 5 and a y-intercept of 10, meaning the line rises 5 units up for each unit it moves right. Understanding linear equations is crucial for solving real-world problems involving constant rates and linear relationships.

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Most popular questions from this chapter

List the potential rational zeros of each polynomial function. Do not attempt to find the zeros. $$ f(x)=3 x^{5}-x^{2}+2 x+18 $$

Is \(\frac{1}{3}\) a zero of \(f(x)=2 x^{3}+3 x^{2}-6 x+7 ?\) Explain.

If \(f(x)=\sqrt{3 x-1}\) and \(g(x)=\sqrt{3 x+1}\) find \((f \cdot g)(x)\) and state its domain.

Use the Factor Theorem to prove that \(x-c\) is a factor of \(x^{n}-c^{n}\) for any positive integer \(n\).

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the vertex of the graph of \(f(x)=3 x^{2}-12 x+7\)
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