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(3) $}$?Select Each Correct Answer.- { -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) and g(x), on the same coordinate plane and find the solutions to the equation f(x) = g(3).
Graphing f(x) = -2x
The first function we will graph is f(x) = -2x. This is a linear function with a slope of -2 and a y-intercept of 0. To graph this function, we can start by plotting the y-intercept, which is the point where the function crosses the y-axis. Since the y-intercept is 0, we can plot the point (0, 0).
Next, we can use the slope to determine the direction and steepness of the line. A negative slope indicates that the line slopes downward from left to right. We can then plot additional points on the line by using the slope to determine the change in y for a given change in x.
Graphing g(x) = x^2 - 3
The second function we will graph is g(x) = x^2 - 3. This is a quadratic function with a leading coefficient of 1 and a constant term of -3. To graph this function, we can start by plotting the vertex, which is the point where the function reaches its maximum or minimum value.
The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a is the leading coefficient and b is the coefficient of the x term. In this case, a = 1 and b = 0, so the x-coordinate of the vertex is x = 0.
The y-coordinate of the vertex can be found by substituting the x-coordinate into the function. In this case, g(0) = 0^2 - 3 = -3, so the vertex is the point (0, -3).
Graphing f(x) and g(x) on the Same Coordinate Plane
Now that we have graphed both functions, we can graph them on the same coordinate plane. To do this, we can superimpose the two graphs on the same set of axes.
The graph of f(x) = -2x is a straight line with a slope of -2 and a y-intercept of 0. The graph of g(x) = x^2 - 3 is a parabola with a vertex at (0, -3).
Finding the Solutions to the Equation f(x) = g(3)
Now that we have graphed both functions on the same coordinate plane, we can find the solutions to the equation f(x) = g(3). To do this, we can substitute x = 3 into the function g(x) = x^2 - 3 to find the value of g(3).
g(3) = 3^2 - 3 = 9 - 3 = 6
Now that we have found the value of g(3), we can substitute it into the equation f(x) = g(3) to find the solutions.
f(x) = g(3) -2x = 6
To solve for x, we can divide both sides of the equation by -2.
x = -6 / 2 x = -3
Therefore, the solution to the equation f(x) = g(3) is x = -3.
Conclusion
Graphing functions on the same coordinate plane is an essential skill in mathematics, particularly in algebra and calculus. By graphing two functions, f(x) and g(x), on the same coordinate plane, we can visualize their behavior, identify their key features, and solve equations. In this article, we graphed f(x) = -2x and g(x) = x^2 - 3 on the same coordinate plane and found the solutions to the equation f(x) = g(3). We hope this article has provided a clear and concise guide to graphing functions on the same coordinate plane.
Additional Resources
For additional resources on graphing functions on the same coordinate plane, including video tutorials and practice problems, please visit the following websites:
- Khan Academy: Graphing Functions
- Mathway: Graphing Functions
- Wolfram Alpha: Graphing Functions
Discussion Questions
- What is the difference between graphing a function and graphing a relation?
- How do you determine the direction and steepness of a line?
- What is the vertex of a parabola, and how do you find it?
- How do you graph a quadratic function on a coordinate plane?
- What is the solution to the equation f(x) = g(3)?
Answer Key
- A function is a relation where each input corresponds to exactly one output, while a relation is a set of ordered pairs.
- The direction and steepness of a line can be determined by the slope, which is the ratio of the change in y to the change in x.
- The vertex of a parabola is the point where the function reaches its maximum or minimum value, and it can be found using the formula x = -b / 2a.
- A quadratic function can be graphed on a coordinate plane by plotting the vertex and using the slope to determine the direction and steepness of the line.
- The solution to the equation f(x) = g(3) is x = -3.
Q&A: Graphing Functions on the Same Coordinate Plane =====================================================
Q: What is the difference between graphing a function and graphing a relation?
A: A function is a relation where each input corresponds to exactly one output, while a relation is a set of ordered pairs. When graphing a function, we need to ensure that each input corresponds to exactly one output, whereas when graphing a relation, we can have multiple outputs for a single input.
Q: How do you determine the direction and steepness of a line?
A: The direction and steepness of a line can be determined by the slope, which is the ratio of the change in y to the change in x. A positive slope indicates that the line slopes upward from left to right, while a negative slope indicates that the line slopes downward from left to right. A steep slope indicates a large change in y for a small change in x, while a shallow slope indicates a small change in y for a large change in x.
Q: What is the vertex of a parabola, and how do you find it?
A: The vertex of a parabola is the point where the function reaches its maximum or minimum value. The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a is the leading coefficient and b is the coefficient of the x term. The y-coordinate of the vertex can be found by substituting the x-coordinate into the function.
Q: How do you graph a quadratic function on a coordinate plane?
A: To graph a quadratic function on a coordinate plane, we can start by plotting the vertex, which is the point where the function reaches its maximum or minimum value. We can then use the slope to determine the direction and steepness of the line. We can also plot additional points on the line by using the slope to determine the change in y for a given change in x.
Q: What is the solution to the equation f(x) = g(3)?
A: To find the solution to the equation f(x) = g(3), we need to substitute x = 3 into the function g(x) = x^2 - 3 to find the value of g(3). We can then substitute this value into the equation f(x) = g(3) to find the solutions.
Q: How do you graph two functions on the same coordinate plane?
A: To graph two functions on the same coordinate plane, we can superimpose the two graphs on the same set of axes. We can then use the graph to visualize the behavior of the 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 using a consistent scale for the axes
- Not labeling the axes correctly
- Not plotting the vertex of a parabola correctly
- Not using the correct slope to determine the direction and steepness of a line
Q: How can I practice graphing functions on the same coordinate plane?
A: There are many resources available to help you practice graphing functions on the same coordinate plane, including:
- Online graphing calculators, such as Desmos or Graphing Calculator
- Practice problems and worksheets, such as those found on Khan Academy or Mathway
- Video tutorials and lessons, such as those found on YouTube or Crash Course
Q: What are some real-world applications of graphing functions on the same coordinate plane?
A: Graphing functions on the same coordinate plane has many real-world applications, including:
- Modeling population growth or decline
- Analyzing the behavior of physical systems, such as springs or pendulums
- Optimizing the design of a product or system
- Predicting the behavior of a complex system, such as a financial market or a weather pattern.
Conclusion
Graphing functions on the same coordinate plane is an essential skill in mathematics, particularly in algebra and calculus. By understanding how to graph functions on the same coordinate plane, we can visualize their behavior, identify their key features, and solve equations. We hope this Q&A article has provided a clear and concise guide to graphing functions on the same coordinate plane.