Find The Coordinates Of The Point(s) Where The Graph Of The Function Intersects The \[$x\$\]-axis.Function: \[$y=\frac{2x-5}{x+3}\$\]Answer: The Coordinates Are \[$(\square, \square)\$\], \[$(\square, \square)\$\].

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Introduction

In mathematics, the intersection of a graph with the x-axis is a crucial concept, especially when dealing with functions. The x-axis is defined by the equation y = 0, and any point on the x-axis has a y-coordinate of 0. To find the coordinates of the point(s) where the graph of a function intersects the x-axis, we need to determine the x-coordinate(s) at which the function's y-value is equal to 0. In this article, we will explore how to find the coordinates of the intersection points with the x-axis for the given function y = (2x - 5)/(x + 3).

Understanding the Function

The given function is y = (2x - 5)/(x + 3). This is a rational function, which means it is the ratio of two polynomials. The numerator is 2x - 5, and the denominator is x + 3. To find the coordinates of the intersection points with the x-axis, we need to set the function equal to 0 and solve for x.

Setting the Function Equal to 0

To find the coordinates of the intersection points with the x-axis, we set the function equal to 0:

y = (2x - 5)/(x + 3) = 0

Solving for x

To solve for x, we need to find the values of x that make the numerator (2x - 5) equal to 0. This is because the denominator (x + 3) cannot be equal to 0, as it would result in an undefined value.

2x - 5 = 0

Solving the Linear Equation

To solve the linear equation 2x - 5 = 0, we can add 5 to both sides of the equation:

2x = 5

Next, we can divide both sides of the equation by 2 to solve for x:

x = 5/2

Finding the Coordinates of the Intersection Points

Now that we have found the x-coordinate of the intersection point, we can find the corresponding y-coordinate by substituting x into the original function:

y = (2(5/2) - 5)/(5/2 + 3)

Simplifying the Expression

To simplify the expression, we can start by evaluating the numerator:

2(5/2) - 5 = 5 - 5 = 0

Next, we can evaluate the denominator:

5/2 + 3 = 5/2 + 6/2 = 11/2

Finding the y-Coordinate

Now that we have simplified the expression, we can find the y-coordinate by substituting the values into the original function:

y = 0/(11/2)

Evaluating the Expression

To evaluate the expression, we can multiply the numerator by the reciprocal of the denominator:

y = 0 × (2/11)

Simplifying the Expression

To simplify the expression, we can multiply the numerator by the reciprocal of the denominator:

y = 0

Finding the Coordinates of the Intersection Points

Now that we have found the y-coordinate, we can find the coordinates of the intersection points by combining the x-coordinate and the y-coordinate:

The coordinates of the intersection points are (5/2, 0).

Conclusion

In this article, we have explored how to find the coordinates of the point(s) where the graph of the function intersects the x-axis. We have used the given function y = (2x - 5)/(x + 3) and set it equal to 0 to find the x-coordinate(s) of the intersection points. We have then found the corresponding y-coordinate(s) by substituting the x-coordinate(s) into the original function. The coordinates of the intersection points are (5/2, 0).

Final Answer

The coordinates of the intersection points are (5/2, 0).

Discussion

The intersection of a graph with the x-axis is an important concept in mathematics, especially when dealing with functions. By setting the function equal to 0 and solving for x, we can find the coordinates of the intersection points. In this article, we have used the given function y = (2x - 5)/(x + 3) to find the coordinates of the intersection points. The coordinates of the intersection points are (5/2, 0).

Related Topics

  • Finding the coordinates of the intersection points with the y-axis
  • Graphing rational functions
  • Solving linear equations
  • Evaluating expressions

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Mathematics for the Nonmathematician" by Morris Kline

Introduction

In our previous article, we explored how to find the coordinates of the point(s) where the graph of the function intersects the x-axis. We used the given function y = (2x - 5)/(x + 3) and set it equal to 0 to find the x-coordinate(s) of the intersection points. In this article, we will answer some frequently asked questions about finding the coordinates of the intersection points with the x-axis.

Q: What is the x-axis?

A: The x-axis is a horizontal line that is defined by the equation y = 0. Any point on the x-axis has a y-coordinate of 0.

Q: How do I find the coordinates of the intersection points with the x-axis?

A: To find the coordinates of the intersection points with the x-axis, you need to set the function equal to 0 and solve for x. Then, you can find the corresponding y-coordinate by substituting x into the original function.

Q: What if the function has multiple intersection points with the x-axis?

A: If the function has multiple intersection points with the x-axis, you will need to find the x-coordinate(s) of each intersection point and then find the corresponding y-coordinate(s) by substituting x into the original function.

Q: How do I know if the function intersects the x-axis at a single point or multiple points?

A: To determine if the function intersects the x-axis at a single point or multiple points, you need to examine the graph of the function. If the graph intersects the x-axis at a single point, then the function intersects the x-axis at a single point. If the graph intersects the x-axis at multiple points, then the function intersects the x-axis at multiple points.

Q: Can I use the same method to find the coordinates of the intersection points with the y-axis?

A: No, you cannot use the same method to find the coordinates of the intersection points with the y-axis. To find the coordinates of the intersection points with the y-axis, you need to set the function equal to x and solve for y.

Q: What if the function has a vertical asymptote at x = a?

A: If the function has a vertical asymptote at x = a, then the function intersects the x-axis at x = a. However, the function is undefined at x = a, so you will need to use a different method to find the coordinates of the intersection points.

Q: Can I use the same method to find the coordinates of the intersection points with a horizontal line?

A: No, you cannot use the same method to find the coordinates of the intersection points with a horizontal line. To find the coordinates of the intersection points with a horizontal line, you need to set the function equal to the equation of the horizontal line and solve for x.

Q: What if the function has a hole at x = a?

A: If the function has a hole at x = a, then the function intersects the x-axis at x = a. However, the function is undefined at x = a, so you will need to use a different method to find the coordinates of the intersection points.

Q: Can I use the same method to find the coordinates of the intersection points with a vertical line?

A: No, you cannot use the same method to find the coordinates of the intersection points with a vertical line. To find the coordinates of the intersection points with a vertical line, you need to set the function equal to the equation of the vertical line and solve for y.

Q: What if the function has a slant asymptote?

A: If the function has a slant asymptote, then the function intersects the x-axis at a single point. However, the function is undefined at the point of intersection, so you will need to use a different method to find the coordinates of the intersection points.

Q: Can I use the same method to find the coordinates of the intersection points with a curve?

A: No, you cannot use the same method to find the coordinates of the intersection points with a curve. To find the coordinates of the intersection points with a curve, you need to use a different method, such as finding the points of intersection between the curve and the x-axis.

Conclusion

In this article, we have answered some frequently asked questions about finding the coordinates of the intersection points with the x-axis. We have discussed how to find the coordinates of the intersection points, how to determine if the function intersects the x-axis at a single point or multiple points, and how to handle different types of functions, such as functions with vertical asymptotes, holes, and slant asymptotes.

Final Answer

The coordinates of the intersection points with the x-axis can be found by setting the function equal to 0 and solving for x. The corresponding y-coordinate can be found by substituting x into the original function.

Discussion

Finding the coordinates of the intersection points with the x-axis is an important concept in mathematics, especially when dealing with functions. By understanding how to find the coordinates of the intersection points, you can better analyze and graph functions.

Related Topics

  • Finding the coordinates of the intersection points with the y-axis
  • Graphing rational functions
  • Solving linear equations
  • Evaluating expressions

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by James Stewart
  • [3] "Mathematics for the Nonmathematician" by Morris Kline