The Equation Of A Linear Function In Point-slope Form Is Y − Y 1 = M ( X − X 1 Y - Y_1 = M(x - X_1 Y − Y 1 ​ = M ( X − X 1 ​ ]. Harold Correctly Wrote The Equation Y = − 7 Y = -7 Y = − 7 Using A Point And The Slope. Which Point Did Harold Use?A. (7, 3) B. (0, 7) C. (7, 0) D. (3, 7)

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Understanding the Point-Slope Form

The equation of a linear function in point-slope form is given by y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line. This form is useful when we know the slope of the line and a point on the line, and we want to write the equation of the line.

Harold's Equation

Harold correctly wrote the equation y=−7y = -7 using a point and the slope. To find the point that Harold used, we need to analyze the equation y=−7y = -7. This equation represents a horizontal line that intersects the y-axis at y=−7y = -7. Since the equation is in the form y=cy = c, where cc is a constant, the line is horizontal and has a slope of 0.

Finding the Point

Since the line is horizontal, the x-coordinate of the point that Harold used must be 0. This is because the x-coordinate of any point on a horizontal line is constant. Therefore, the point that Harold used must be of the form (0,y)(0, y), where yy is the y-coordinate of the point.

Analyzing the Options

Now, let's analyze the options given:

A. (7, 3) B. (0, 7) C. (7, 0) D. (3, 7)

Since the x-coordinate of the point that Harold used must be 0, options A, C, and D can be eliminated. This leaves us with option B, (0, 7).

Conclusion

Therefore, the point that Harold used is (0, 7).

The Importance of Understanding Point-Slope Form

Understanding the point-slope form of a linear function is crucial in mathematics, particularly in algebra and geometry. It allows us to write the equation of a line when we know the slope and a point on the line. This form is also useful in solving problems involving linear equations and inequalities.

Real-World Applications

The point-slope form has numerous real-world applications, including:

  • Physics: The point-slope form is used to describe the motion of objects under constant acceleration.
  • Engineering: The point-slope form is used to design and analyze the performance of mechanical systems, such as bridges and buildings.
  • Computer Science: The point-slope form is used in computer graphics to create and manipulate 2D and 3D objects.

Conclusion

In conclusion, the point-slope form of a linear function is a powerful tool in mathematics, particularly in algebra and geometry. Understanding this form is crucial in solving problems involving linear equations and inequalities, and it has numerous real-world applications in physics, engineering, and computer science.

Additional Resources

For further reading and practice, we recommend the following resources:

  • Algebra textbooks: "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart.
  • Online resources: Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
  • Practice problems: IXL, Mathway, and Symbolab.

Final Thoughts

Frequently Asked Questions

Q: What is the point-slope form of a linear function? A: The point-slope form of a linear function is given by y−y1=m(x−x1)y - y_1 = m(x - x_1), where mm is the slope of the line and (x1,y1)(x_1, y_1) is a point on the line.

Q: How do I use the point-slope form to write the equation of a line? A: To use the point-slope form, you need to know the slope of the line and a point on the line. Then, you can plug in the values of the slope and the point into the equation y−y1=m(x−x1)y - y_1 = m(x - x_1).

Q: What is the significance of the point (x1,y1)(x_1, y_1) in the point-slope form? A: The point (x1,y1)(x_1, y_1) is a point on the line, and it is used to determine the equation of the line. The coordinates of this point are used to calculate the slope of the line.

Q: Can I use the point-slope form to find the equation of a horizontal line? A: Yes, you can use the point-slope form to find the equation of a horizontal line. Since the slope of a horizontal line is 0, you can plug in 0 for the slope in the equation y−y1=m(x−x1)y - y_1 = m(x - x_1).

Q: Can I use the point-slope form to find the equation of a vertical line? A: Yes, you can use the point-slope form to find the equation of a vertical line. Since the slope of a vertical line is undefined, you can plug in a value of 0 for the slope in the equation y−y1=m(x−x1)y - y_1 = m(x - x_1).

Q: How do I determine the slope of a line using the point-slope form? A: To determine the slope of a line using the point-slope form, you need to know the coordinates of two points on the line. Then, you can use the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} to calculate the slope.

Q: Can I use the point-slope form to solve systems of linear equations? A: Yes, you can use the point-slope form to solve systems of linear equations. By using the point-slope form to write the equations of the lines, you can then solve the system of equations using substitution or elimination.

Q: What are some real-world applications of the point-slope form? A: The point-slope form has numerous real-world applications, including physics, engineering, and computer science. It is used to describe the motion of objects under constant acceleration, design and analyze the performance of mechanical systems, and create and manipulate 2D and 3D objects.

Q: How do I practice using the point-slope form? A: You can practice using the point-slope form by working through examples and exercises in your textbook or online resources. You can also try solving problems involving linear equations and inequalities using the point-slope form.

Conclusion

The point-slope form of a linear function is a powerful tool in mathematics, particularly in algebra and geometry. By understanding this form, you can solve problems involving linear equations and inequalities, and you will be well-prepared for the challenges of mathematics and science.