Graph The Functions On The Same Coordinate Plane.${ \begin{array}{l} f(x) = -2x \ g(x) = X^2 - 3 \end{array} }$What Are The Solutions To The Equation F ( X ) = G ( X F(x) = G(x F ( X ) = G ( X ]?Select Each Correct Answer.A. -3 B. -2 C. 0 D. 1 E. 3

Introduction

Graphing functions on the same coordinate plane is an essential skill in mathematics, particularly in algebra and calculus. It allows us to visualize the behavior of functions, identify their key features, and solve equations. In this article, we will explore how to graph two functions, $f(x) = -2x$ and $g(x) = x^2 - 3$, on the same coordinate plane and find the solutions to the equation $f(x) = g(x)$.

Graphing the Functions

To graph the functions, we need to understand their behavior and key features. The function $f(x) = -2x$ is a linear function with a negative slope. This means that as $x$ increases, $f(x)$ decreases. The function $g(x) = x^2 - 3$ is a quadratic function with a positive leading coefficient. This means that the parabola opens upwards, and the vertex is the minimum point.

Graphing $f(x) = -2x$

To graph $f(x) = -2x$, we can start by finding the $y$-intercept, which is the point where the graph intersects the $y$-axis. We can do this by substituting $x = 0$ into the equation:

$$f(0) = -2(0) = 0$$

So, the $y$-intercept is $(0, 0)$. Next, we can find the $x$-intercept, which is the point where the graph intersects the $x$-axis. We can do this by substituting $y = 0$ into the equation:

$$0 = -2x$$

Solving for $x$, we get:

$$x = 0$$

So, the $x$-intercept is $(0, 0)$. Now, we can use the slope-intercept form of a linear equation, $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. In this case, $m = -2$ and $b = 0$. So, the equation becomes:

$$y = -2x$$

We can graph this equation by plotting points on the coordinate plane. For example, we can plot the points $(1, -2)$, $(2, -4)$, and $(3, -6)$.

Graphing $g(x) = x^2 - 3$

To graph $g(x) = x^2 - 3$, we can start by finding the vertex, which is the minimum point of the parabola. We can do this by completing the square:

$$g(x) = x^2 - 3 = (x - 0)^2 - 3$$

So, the vertex is $(0, -3)$. Next, we can find the $y$-intercept, which is the point where the graph intersects the $y$-axis. We can do this by substituting $x = 0$ into the equation:

$$g(0) = (0)^2 - 3 = -3$$

So, the $y$-intercept is $(0, -3)$. Now, we can use the factored form of a quadratic equation, $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In this case, $a = 1$, $h = 0$, and $k = -3$. So, the equation becomes:

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

We can graph this equation by plotting points on the coordinate plane. For example, we can plot the points $(1, -2)$, $(2, -1)$, and $(3, 0)$.

Graphing the Functions on the Same Coordinate Plane

Now that we have graphed the individual functions, we can graph them on the same coordinate plane. To do this, we need to find the intersection points of the two graphs. We can do this by setting the two equations equal to each other and solving for $x$:

$$-2x = x^2 - 3$$

Rearranging the equation, we get:

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

Factoring the equation, we get:

$$(x + 3)(x - 1) = 0$$

Solving for $x$, we get:

$$x = -3 \text{ or } x = 1$$

Now, we can substitute these values of $x$ into one of the original equations to find the corresponding values of $y$. For example, we can substitute $x = -3$ into the equation $f(x) = -2x$:

$$f(-3) = -2(-3) = 6$$

So, the point of intersection is $(-3, 6)$. Similarly, we can substitute $x = 1$ into the equation $f(x) = -2x$:

$$f(1) = -2(1) = -2$$

So, the point of intersection is $(1, -2)$.

Solutions to the Equation $f(x) = g(x)$

The solutions to the equation $f(x) = g(x)$ are the points of intersection of the two graphs. In this case, the solutions are $(-3, 6)$ and $(1, -2)$.

Conclusion

Graphing functions on the same coordinate plane is an essential skill in mathematics. By graphing the functions $f(x) = -2x$ and $g(x) = x^2 - 3$, we can visualize their behavior and key features. We can also find the solutions to the equation $f(x) = g(x)$ by graphing the functions on the same coordinate plane and finding the points of intersection. In this article, we have explored how to graph the functions and find the solutions to the equation. We have also discussed the importance of graphing functions on the same coordinate plane in mathematics.

Final Answer

The final answer is:

• -3
• 1
  Graphing Functions on the Same Coordinate Plane: A Q&A Guide ===========================================================

Introduction

Graphing functions on the same coordinate plane is an essential skill in mathematics, particularly in algebra and calculus. In our previous article, we explored how to graph two functions, $f(x) = -2x$ and $g(x) = x^2 - 3$, on the same coordinate plane and find the solutions to the equation $f(x) = g(x)$. In this article, we will answer some common questions related to graphing functions on the same coordinate plane.

Q&A

Q: What is the purpose of graphing functions on the same coordinate plane?

A: The purpose of graphing functions on the same coordinate plane is to visualize the behavior of functions, identify their key features, and solve equations.

Q: How do I graph a linear function on the same coordinate plane?

A: To graph a linear function on the same coordinate plane, you need to find the $y$-intercept and the $x$-intercept. You can do this by substituting $x = 0$ into the equation to find the $y$-intercept, and by substituting $y = 0$ into the equation to find the $x$-intercept.

Q: How do I graph a quadratic function on the same coordinate plane?

A: To graph a quadratic function on the same coordinate plane, you need to find the vertex, which is the minimum point of the parabola. You can do this by completing the square or by using the factored form of a quadratic equation.

Q: How do I find the intersection points of two graphs on the same coordinate plane?

A: To find the intersection points of two graphs on the same coordinate plane, you need to set the two equations equal to each other and solve for $x$. You can then substitute the values of $x$ into one of the original equations to find the corresponding values of $y$.

Q: What are the solutions to the equation $f(x) = g(x)$?

A: The solutions to the equation $f(x) = g(x)$ are the points of intersection of the two graphs. In this case, the solutions are $(-3, 6)$ and $(1, -2)$.

Q: How do I graph functions on the same coordinate plane using technology?

A: You can graph functions on the same coordinate plane using technology such as graphing calculators or computer software. These tools can help you visualize the behavior of functions and identify their key features.

Q: What are some common mistakes to avoid when graphing functions on the same coordinate plane?

A: Some common mistakes to avoid when graphing functions on the same coordinate plane include:

• Not finding the $y$-intercept and the $x$-intercept of the linear function
• Not finding the vertex of the quadratic function
• Not setting the two equations equal to each other and solving for $x$
• Not substituting the values of $x$ into one of the original equations to find the corresponding values of $y$

Conclusion

Graphing functions on the same coordinate plane is an essential skill in mathematics. By graphing the functions $f(x) = -2x$ and $g(x) = x^2 - 3$, we can visualize their behavior and key features. We can also find the solutions to the equation $f(x) = g(x)$ by graphing the functions on the same coordinate plane and finding the points of intersection. In this article, we have answered some common questions related to graphing functions on the same coordinate plane.

Final Answer

The final answer is:

• Graphing functions on the same coordinate plane is an essential skill in mathematics.
• To graph a linear function, find the $y$-intercept and the $x$-intercept.
• To graph a quadratic function, find the vertex.
• To find the intersection points of two graphs, set the two equations equal to each other and solve for $x$.
• The solutions to the equation $f(x) = g(x)$ are the points of intersection of the two graphs.