Given The Function $f(x) = 3x - 8$,Step 2 Of 2: Find The Y-intercept Of The Line Represented By The Linear Equation. Write Your Answer As A Coordinate Pair.Answer:

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Introduction

In mathematics, a linear equation is a type of equation in which the highest power of the variable(s) is 1. It is often represented in the form of y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is a crucial point on the graph of a linear equation, as it represents the point where the line intersects the y-axis. In this article, we will explore how to find the y-intercept of a linear equation, using the function f(x) = 3x - 8 as an example.

What is the Y-Intercept?

The y-intercept is the point on the graph of a linear equation where the line intersects the y-axis. It is the value of y when x is equal to 0. In other words, it is the point on the graph where the line crosses the y-axis. The y-intercept is denoted by the letter b in the equation y = mx + b.

Finding the Y-Intercept of a Linear Equation

To find the y-intercept of a linear equation, we need to substitute x = 0 into the equation and solve for y. This is because the y-intercept is the point on the graph where the line intersects the y-axis, and at this point, the value of x is 0.

Step-by-Step Solution

Let's use the function f(x) = 3x - 8 as an example. To find the y-intercept, we need to substitute x = 0 into the equation and solve for y.

Step 1: Substitute x = 0 into the Equation

f(x) = 3x - 8 f(0) = 3(0) - 8

Step 2: Simplify the Equation

f(0) = 3(0) - 8 f(0) = 0 - 8 f(0) = -8

Step 3: Write the Y-Intercept as a Coordinate Pair

The y-intercept is the point on the graph where the line intersects the y-axis. Therefore, the y-intercept can be written as a coordinate pair (0, y). In this case, the y-intercept is (0, -8).

Conclusion

In conclusion, the y-intercept of a linear equation is the point on the graph where the line intersects the y-axis. It is the value of y when x is equal to 0. To find the y-intercept, we need to substitute x = 0 into the equation and solve for y. Using the function f(x) = 3x - 8 as an example, we found that the y-intercept is (0, -8).

Example Problems

Problem 1

Find the y-intercept of the linear equation f(x) = 2x + 5.

Solution

f(x) = 2x + 5 f(0) = 2(0) + 5 f(0) = 0 + 5 f(0) = 5 The y-intercept is (0, 5).

Problem 2

Find the y-intercept of the linear equation f(x) = x - 3.

Solution

f(x) = x - 3 f(0) = 0 - 3 f(0) = -3 The y-intercept is (0, -3).

Tips and Tricks

  • To find the y-intercept of a linear equation, substitute x = 0 into the equation and solve for y.
  • The y-intercept is the point on the graph where the line intersects the y-axis.
  • The y-intercept can be written as a coordinate pair (0, y).

Common Mistakes

  • Failing to substitute x = 0 into the equation.
  • Not solving for y after substituting x = 0 into the equation.
  • Writing the y-intercept as a coordinate pair (x, y) instead of (0, y).

Real-World Applications

  • Finding the y-intercept of a linear equation is crucial in many real-world applications, such as:
  • Calculating the cost of a product based on its price and quantity.
  • Determining the amount of money in a bank account based on the interest rate and time.
  • Finding the height of a building based on its shadow and the angle of the sun.

Conclusion

Q: What is the y-intercept of a linear equation?

A: The y-intercept of a linear equation is the point on the graph where the line intersects the y-axis. It is the value of y when x is equal to 0.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept of a linear equation, substitute x = 0 into the equation and solve for y.

Q: What is the formula for finding the y-intercept of a linear equation?

A: The formula for finding the y-intercept of a linear equation is:

y = mx + b

where m is the slope and b is the y-intercept.

Q: How do I write the y-intercept as a coordinate pair?

A: The y-intercept can be written as a coordinate pair (0, y).

Q: What is the difference between the y-intercept and the x-intercept?

A: The y-intercept is the point on the graph where the line intersects the y-axis, while the x-intercept is the point on the graph where the line intersects the x-axis.

Q: Can I find the y-intercept of a linear equation if I don't know the slope?

A: Yes, you can find the y-intercept of a linear equation even if you don't know the slope. Simply substitute x = 0 into the equation and solve for y.

Q: How do I find the y-intercept of a linear equation with a negative slope?

A: To find the y-intercept of a linear equation with a negative slope, substitute x = 0 into the equation and solve for y. The y-intercept will be a negative value.

Q: Can I use a graphing calculator to find the y-intercept of a linear equation?

A: Yes, you can use a graphing calculator to find the y-intercept of a linear equation. Simply graph the equation and use the calculator to find the point where the line intersects the y-axis.

Q: How do I find the y-intercept of a linear equation with a fractional slope?

A: To find the y-intercept of a linear equation with a fractional slope, substitute x = 0 into the equation and solve for y. The y-intercept will be a fraction.

Q: Can I find the y-intercept of a linear equation with a negative fractional slope?

A: Yes, you can find the y-intercept of a linear equation with a negative fractional slope. Simply substitute x = 0 into the equation and solve for y. The y-intercept will be a negative fraction.

Q: How do I find the y-intercept of a linear equation with a decimal slope?

A: To find the y-intercept of a linear equation with a decimal slope, substitute x = 0 into the equation and solve for y. The y-intercept will be a decimal value.

Q: Can I use a graphing calculator to find the y-intercept of a linear equation with a decimal slope?

A: Yes, you can use a graphing calculator to find the y-intercept of a linear equation with a decimal slope. Simply graph the equation and use the calculator to find the point where the line intersects the y-axis.

Q: How do I find the y-intercept of a linear equation with a negative decimal slope?

A: To find the y-intercept of a linear equation with a negative decimal slope, substitute x = 0 into the equation and solve for y. The y-intercept will be a negative decimal value.

Q: Can I use a graphing calculator to find the y-intercept of a linear equation with a negative decimal slope?

A: Yes, you can use a graphing calculator to find the y-intercept of a linear equation with a negative decimal slope. Simply graph the equation and use the calculator to find the point where the line intersects the y-axis.

Conclusion

In conclusion, finding the y-intercept of a linear equation is a crucial concept in mathematics. By following the step-by-step solution outlined in this article, you can find the y-intercept of any linear equation. Remember to substitute x = 0 into the equation and solve for y, and always write the y-intercept as a coordinate pair (0, y).