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Chapter 9: Problem 56

For Exercises \(41-60\), find the coordinates of the \(x\) - and \(y\) -intercepts. $$y=4$$

Short Answer

Expert verified
Answer: There is no x-intercept, and the y-intercept is at the point $$(0, 4)$$.

Step by step solution

01

Write down the given equation

We are given the equation $$y = 4$$.
02

Find the x-intercept

To find the x-intercept, we need to set y equal to 0 in the equation and solve for x. $$0 = 4$$ Since this equation is not possible (there is no value of x that could make it work), there is no x-intercept for the given equation.
03

Find the y-intercept

To find the y-intercept, we need to set x equal to 0 in the equation and solve for y. However, our equation does not contain any x variable, so the equation remains: $$y = 4$$ The y-intercept is simply at $$y = 4$$.
04

Write down the intercepts

We found that there is no x-intercept and the y-intercept is at $$y = 4$$ or the point $$(0, 4)$$.

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Key Concepts

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

x-intercept
The **x-intercept** of a function is the point where the graph of the equation crosses or touches the x-axis. This can be effectively understood as the value of \(x\) when \(y = 0\).
In many equations, determining the x-intercept is as simple as setting the value of \(y\) in the equation to zero and solving for \(x\). However, for the equation \(y = 4\), there is no \(x\) variable to manipulate.
  • When there are no possible solutions during this set-up, it indicates that there is no actual x-intercept.
  • This happens because the line described by the equation never crosses the x-axis.
In this specific case, the linear equation \(y = 4\) is a horizontal line parallel to the x-axis, located four units above it. Therefore, it never intersects the x-axis, confirming that there is no x-intercept.
y-intercept
The **y-intercept** is the point at which a graph intersects the y-axis of the coordinate system. Essentially, this is where \(x\) is set to zero.
  • For linear equations, the y-intercept can often be read directly from the equation itself.
  • When the equation is in the format \(y = mx + b\), the term \(b\) is often the y-intercept.
In the current equation, \(y = 4\), the slope form is slightly different since there's no \(x\) term present. This directly tells us that at \(x = 0\), \(y\) remains 4. Therefore, the y-intercept is simply the point \((0, 4)\).
Knowing that the graph will cross the y-axis at this point tells us the coordinate of the intercept without further calculation. Thus, when dealing with such simple horizontal lines, finding the y-intercept becomes straightforward.
coordinate system
Understanding a **coordinate system** is crucial for determining intercepts in any graph. A coordinate system is a grid formed by intersecting lines called axes, which are usually perpendicular to each other.
  • The horizontal axis is referred to as the x-axis.
  • The vertical axis is termed as the y-axis.
  • Any point in this system is represented by an ordered pair \((x, y)\).
This ordered pair shows the exact location of that point on the plane.
  • The x-value indicates horizontal movement from the origin.
  • The y-value indicates vertical movement.
Intercepts are related to where a graph meets these axes, giving insight into the function's behavior as \(x\) or \(y\) approaches zero. A clear grasp of the coordinate system allows students to confidently map out graphs and understand various algebraic concepts.

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Most popular questions from this chapter

For Exercises \(13-36,\) find three solutions for the given equation and graph. (Answers may vary for the three solutions.) $$y=-2 x+4$$

Determine the quadrant in which the point is located. $$(42,-60)$$

For Exercises \(41-60\), find the coordinates of the \(x\) - and \(y\) -intercepts. $$\frac{3}{4} x-y=-6$$

Find the midpoint of the given points. $$(1,7) \text { and }(9,-8)$$

Use the given linear equation to answer the questions. The equation \(C=\frac{5}{9}(F-32)\) is used to convert a temperature in \(^{\circ} \mathrm{F}\) to temperature in \(^{\circ} \mathrm{C}.\) a. What is the \(F\) -intercept? b. What is the C-intercept? c. Convert \(68^{\circ} \mathrm{F}\) to \(^{\circ} \mathrm{C}\). d. Graph the equation with \(F\) on the horizontal axis and C on the vertical axis.
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