Determine The \[$x\$\]- And \[$y\$\]-intercepts For The Graph Defined By The Given Equation: $\[y = X + 8\\]a. \[$x\$\]-intercept Is \[$(0, 8)\$\]; \[$y\$\]-intercept Is \[$(-8, 0)\$\]b.

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Introduction

In mathematics, the intercepts of a graph are the points where the graph intersects the x-axis and the y-axis. The $x$-intercept is the point where the graph intersects the x-axis, and the $y$-intercept is the point where the graph intersects the y-axis. In this article, we will determine the $x$- and $y$-intercepts for the graph defined by the given equation $y = x + 8$.

Understanding the Equation

The given equation is a linear equation in the form of $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. In this case, the equation is $y = x + 8$, where the slope is $1$ and the y-intercept is $8$. This means that the graph of the equation will be a straight line with a slope of $1$ and a y-intercept of $8$.

Finding the $x$-Intercept

To find the $x$-intercept, we need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. We can substitute $y = 0$ into the equation $y = x + 8$ and solve for $x$.

$$0 = x + 8$$

Subtracting $8$ from both sides gives us:

$$-8 = x$$

Therefore, the $x$-intercept is $(-8, 0)$.

Finding the $y$-Intercept

To find the $y$-intercept, we need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. We can substitute $x = 0$ into the equation $y = x + 8$ and solve for $y$.

$$y = 0 + 8$$

Simplifying the equation gives us:

$$y = 8$$

Therefore, the $y$-intercept is $(0, 8)$.

Conclusion

In conclusion, we have determined the $x$- and $y$-intercepts for the graph defined by the given equation $y = x + 8$. The $x$-intercept is $(-8, 0)$, and the $y$-intercept is $(0, 8)$. These intercepts are important in understanding the graph of the equation and can be used to determine the behavior of the graph.

Example Problems

Problem 1

Find the $x$- and $y$-intercepts for the graph defined by the equation $y = 2x - 3$.

Solution

To find the $x$-intercept, we need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. We can substitute $y = 0$ into the equation $y = 2x - 3$ and solve for $x$.

$$0 = 2x - 3$$

Adding $3$ to both sides gives us:

$$3 = 2x$$

Dividing both sides by $2$ gives us:

$$\frac{3}{2} = x$$

Therefore, the $x$-intercept is $(\frac{3}{2}, 0)$.

To find the $y$-intercept, we need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. We can substitute $x = 0$ into the equation $y = 2x - 3$ and solve for $y$.

$$y = 2(0) - 3$$

Simplifying the equation gives us:

$$y = -3$$

Therefore, the $y$-intercept is $(0, -3)$.

Problem 2

Find the $x$- and $y$-intercepts for the graph defined by the equation $y = -x + 5$.

Solution

To find the $x$-intercept, we need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. We can substitute $y = 0$ into the equation $y = -x + 5$ and solve for $x$.

$$0 = -x + 5$$

Adding $x$ to both sides gives us:

$$x = 5$$

Therefore, the $x$-intercept is $(5, 0)$.

To find the $y$-intercept, we need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. We can substitute $x = 0$ into the equation $y = -x + 5$ and solve for $y$.

$$y = -0 + 5$$

Simplifying the equation gives us:

$$y = 5$$

Therefore, the $y$-intercept is $(0, 5)$.

Key Takeaways

• The $x$-intercept is the point where the graph intersects the x-axis.
• The $y$-intercept is the point where the graph intersects the y-axis.
• To find the $x$-intercept, we need to find the point where the graph intersects the x-axis by substituting $y = 0$ into the equation.
• To find the $y$-intercept, we need to find the point where the graph intersects the y-axis by substituting $x = 0$ into the equation.

Final Thoughts

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Introduction

In our previous article, we discussed how to determine the $x$- and $y$-intercepts for a graph defined by a linear equation. In this article, we will provide a Q&A section to help clarify any questions or doubts you may have about the concept.

Q&A

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

A: The $x$-intercept is the point where the graph intersects the x-axis, and the $y$-intercept is the point where the graph intersects the y-axis.

Q: How do I find the $x$-intercept of a graph?

A: To find the $x$-intercept, you need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. You can substitute $y = 0$ into the equation and solve for $x$.

Q: How do I find the $y$-intercept of a graph?

A: To find the $y$-intercept, you need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. You can substitute $x = 0$ into the equation and solve for $y$.

Q: What if the equation is not in the form $y = mx + b$?

A: If the equation is not in the form $y = mx + b$, you can still find the $x$- and $y$-intercepts by using the same methods. However, you may need to rearrange the equation to isolate $y$ or $x$.

Q: Can I find the $x$- and $y$-intercepts of a graph if it is not a linear equation?

A: Yes, you can find the $x$- and $y$-intercepts of a graph even if it is not a linear equation. However, the methods for finding the intercepts may be more complex and may require the use of calculus or other advanced mathematical techniques.

Q: How do I use the $x$- and $y$-intercepts to understand the behavior of a graph?

A: The $x$- and $y$-intercepts can be used to understand the behavior of a graph by providing information about the graph's asymptotes, intercepts, and other key features.

Q: Can I use the $x$- and $y$-intercepts to make predictions about the behavior of a graph?

A: Yes, you can use the $x$- and $y$-intercepts to make predictions about the behavior of a graph. By understanding the graph's intercepts and other key features, you can make informed predictions about the graph's behavior.

Example Problems

Problem 1

Find the $x$- and $y$-intercepts for the graph defined by the equation $y = 2x^2 + 3x - 4$.

Solution

To find the $x$-intercept, we need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. We can substitute $y = 0$ into the equation and solve for $x$.

$$0 = 2x^2 + 3x - 4$$

Using the quadratic formula, we get:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-4)}}{2(2)}$$

Simplifying the equation gives us:

$$x = \frac{-3 \pm \sqrt{9 + 32}}{4}$$

$$x = \frac{-3 \pm \sqrt{41}}{4}$$

Therefore, the $x$-intercepts are $\left(\frac{-3 + \sqrt{41}}{4}, 0\right)$ and $\left(\frac{-3 - \sqrt{41}}{4}, 0\right)$.

To find the $y$-intercept, we need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. We can substitute $x = 0$ into the equation and solve for $y$.

$$y = 2(0)^2 + 3(0) - 4$$

Simplifying the equation gives us:

$$y = -4$$

Therefore, the $y$-intercept is $(0, -4)$.

Problem 2

Find the $x$- and $y$-intercepts for the graph defined by the equation $y = \frac{1}{x} + 2$.

Solution

To find the $x$-intercept, we need to find the point where the graph intersects the x-axis. This occurs when $y = 0$. We can substitute $y = 0$ into the equation and solve for $x$.

$$0 = \frac{1}{x} + 2$$

Subtracting $2$ from both sides gives us:

$$-2 = \frac{1}{x}$$

Multiplying both sides by $x$ gives us:

$$-2x = 1$$

Dividing both sides by $-2$ gives us:

$$x = -\frac{1}{2}$$

Therefore, the $x$-intercept is $\left(-\frac{1}{2}, 0\right)$.

To find the $y$-intercept, we need to find the point where the graph intersects the y-axis. This occurs when $x = 0$. We can substitute $x = 0$ into the equation and solve for $y$.

$$y = \frac{1}{0} + 2$$

This equation is undefined, so we cannot find the $y$-intercept.

Key Takeaways

• The $x$-intercept is the point where the graph intersects the x-axis.
• The $y$-intercept is the point where the graph intersects the y-axis.
• To find the $x$-intercept, you need to find the point where the graph intersects the x-axis by substituting $y = 0$ into the equation.
• To find the $y$-intercept, you need to find the point where the graph intersects the y-axis by substituting $x = 0$ into the equation.

Final Thoughts

In conclusion, the $x$- and $y$-intercepts are important concepts in mathematics that can be used to understand the behavior of a graph. By understanding how to find the intercepts, you can make predictions about the graph's behavior and use the intercepts to make informed decisions. In this article, we have provided a Q&A section to help clarify any questions or doubts you may have about the concept. We have also provided example problems to help illustrate the concept.