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Chapter 2: Problem 5

Find the \(y\) -intercept of the graph of each equation. a. \(y=2 x\) b. \(x=-3\)

Short Answer

Expert verified
a. \((0, 0)\) b. No y-intercept

Step by step solution

01

Understand the Concept

The y-intercept of a graph is the point where the graph crosses the y-axis. At this point, the value of x is 0. We need to substitute x=0 into the given equations to find the corresponding value of y.
02

Solve Part (a)

For the equation \(y=2x\), substitute \(x=0\) into the equation: \[y=2(0)\]\[y=0\]Thus, the y-intercept for this equation is the point \((0, 0)\).
03

Solve Part (b)

The equation \(x=-3\) is a vertical line that runs through \(x=-3\). Vertical lines do not cross the y-axis, so technically it does not have a y-intercept. In cases of undefined intercepts, we consider there is no y-intercept for this line.

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Key Concepts

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

Understanding Linear Equations
Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. A typical linear equation is written in the form \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

This form is known as the slope-intercept form because it easily shows the slope and the y-intercept of the line.
  • The slope, \(m\), indicates how steep the line is.
  • The y-intercept, \(b\), is the point where the line crosses the y-axis, and it occurs when \(x = 0\).
This understanding helps us easily graph linear equations or find specific features such as the y-intercept, as seen in the equation example of \(y = 2x\), where substituting \(x = 0\) gives the y-intercept at \((0,0)\). Being familiar with various forms of linear equations is important when solving for different unknowns or graphing lines.
Graphing Linear Equations
Graphing linear equations involves representing algebraic equations as lines on a coordinate grid. Here鈥檚 how you can start:
  • Identify the equation's form: Make sure your equation is in or can easily be converted into the slope-intercept form, \(y = mx + b\).
  • Find key points: Use the y-intercept \((0, b)\) as a starting point. For example, in the equation \(y = 2x\), the y-intercept is \((0,0)\).
  • Calculate additional points: Choose other values of \(x\) to substitute into the equation to find corresponding \(y\) values.
  • Draw the line: Plot the points on the graph, and draw a straight line through them.
Remember, every point on this line is a solution to the equation \(y=2x\), demonstrating the consistent increase or decrease in y with changes in x according to the slope.
The Nature of a Vertical Line
A vertical line on a graph represents a unique condition for linear equations where the x-value remains constant, depicted by an equation like \(x = -3\). For vertical lines:
  • The x-coordinate for all points on the line is the same.
  • Such lines run parallel to the y-axis.
  • They never cross the y-axis unless at infinity, which is why they are considered to have no y-intercept.
Understanding a vertical line鈥檚 behavior is crucial as it highlights a departure from typical linear equations, where both x and y vary. In our example with \(x = -3\), all points on this line will have an x-coordinate of -3, but a y-coordinate that can be any value.

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Most popular questions from this chapter

Find \(h(5)\) and \(h(-2)\). \(h(x)=\frac{x}{x^{2}+2}\)

Write an equation for a linear function whose graph has the given characteristics. Passes through \((2,20),\) parallel to the graph of \(g(x)=8 x+1\)

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples 2, 3, and 4. $$ f(x)=(x-1)^{3} $$

a. Suppose you know the slope of a line. Is that enough information about the line to write its equation? Explain. b. Suppose you know the coordinates of a point on a line. Is that enough information about the line to write its equation? Explain.

Find \(g(2)\) and \(g(3)\). \(g(x)=(x-3)^{2}\)
See all solutions

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