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Chapter 2: Problem 6

An \(x\) -intercept of a graph has a \(y\) -coordinate of _____.

Short Answer

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Step by step solution

01

Identify the Concept

An x-intercept is a point where the graph of a function crosses the x-axis.
02

Determine the Characteristics

At any x-intercept, the y-coordinate is always 0 because the point lies on the x-axis.
03

Apply the Information

Since the problem is asking for the y-coordinate of an x-intercept, we can conclude that the y-coordinate is 0.

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Key Concepts

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

coordinate system
To understand the concept of x-intercepts, it's crucial to be familiar with the coordinate system. The coordinate system is a two-dimensional system used to graph points, lines, and curves. It consists of two axes: the x-axis (horizontal) and the y-axis (vertical).

The point where these axes intersect is called the origin, noted as \( (0,0) \).

This system allows you to locate any point in a plane using an ordered pair \( (x,y) \), where \( x \) indicates the horizontal position and \( y \) the vertical position.

Understanding this system is key to graphing functions and finding x-intercepts.
graphing functions
Graphing functions is a way to visualize equations and understand the relationships between variables. When you graph a function, you plot points that satisfy the equation.
For example, when graphing \( y = 2x + 1 \), you calculate points such as \( (0,1) \) and \( (1,3) \), plot them on the coordinate system, and draw a line through them.

Graphing helps to see where the function intersects the axes, which can be very useful for solving problems about x-intercepts.

Understanding the intersection points can give insights into solutions of equations and the behavior of functions.
x-axis
The x-axis is a fundamental part of the coordinate system. It runs horizontally and is used as a reference line for measuring vertical distances (y-values).

Any point on the x-axis has a y-coordinate of 0 because it does not move up or down from the origin. This is essential when identifying x-intercepts.

An x-intercept is specifically where a graph crosses the x-axis so, by definition, the y-coordinate at these points is always 0.

This means that to find an x-intercept, you only need to look where the function touches or crosses the x-axis.

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Most popular questions from this chapter

Evaluate the function for the given values of \(x\). \(h(x)=\left\\{\begin{aligned} 2 & \text { for }-3 \leq x<-2 \\ 1 & \text { for }-2 \leq x<-1 \\ 0 & \text { for }-1 \leq x<0 \\\\-1 & \text { for } 0 \leq x<1 \end{aligned}\right.\) a. \(h(-1.7)\) b. \(h(-2.5)\) c. \(h(0.05)\) d. \(h(-2)\) \(\mathbf{e} . h(0)\)

The water level in a retention pond started at \(5 \mathrm{ft}(60 \mathrm{in} .)\) and decreased at a rate of 2 in./day during a 14 -day drought. A tropical depression moved through at the beginning of the 15 th day and produced rain at an average rate of 2.5 in./day for 5 days. a. Write a piecewise-defined function to model the water level \(L(x)\) (in inches) as a function of the number of days \(x\) since the beginning of the drought. b. Graph the function.

A function is given. (See Examples \(4-5)\) a. Find \(f(x+h)\). b. Find \(\frac{f(x+h)-f(x)}{h}\). $$f(x)=x^{2}+4 x$$

A website designer creates videos on how to create websites. She sells the videos in 10 -hr packages for \(\$ 40\) each. Her one-time initial cost to produce each 10 -hr video package is \(\$ 5000\) (this includes labor and the cost of computer supplies). The cost to package and ship each \(\mathrm{CD}\) is \(\$ 2.80\). a. Write a linear cost function that represents the \(\operatorname{cost} C(x)\) to produce, package, and ship \(x\) 10-hr video packages. b. Write a linear revenue function to represent the revenue \(R(x)\) for selling \(x\) 10-hr video packages. c. Evaluate \((R-C)(x)\) and interpret its meaning in the context of this problem. d. Determine the profit if the website designer produces and sells 2400 video packages in the course of one year.

Refer to the functions \(f, g,\) and \(h\) and evaluate the given functions. \(f(x)=2 x+1 \quad g(x)=x^{2} \quad h(x)=\sqrt[3]{x}\) $$(g \circ h \circ f)(x)$$
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