Find A Formula For A Function $g(x)$ Whose Graph Is Obtained From $f(x)=x^2$ By Shifting Right 1 Unit, Vertically Stretching By A Factor Of 4, And Shifting Down 7 Units.\$g(x)=$[/tex\] $\square$
Introduction
In mathematics, functions are used to describe relationships between variables. When working with functions, it's often necessary to transform them in various ways to better understand their behavior or to create new functions that meet specific requirements. In this article, we'll explore the process of shifting, stretching, and reflecting graphs of functions, and we'll use the function $f(x) = x^2$ as a starting point to demonstrate these transformations.
Shifting Functions
Shifting a function involves moving its graph to a new position on the coordinate plane. There are three types of shifts: horizontal, vertical, and a combination of both.
Horizontal Shifts
A horizontal shift occurs when a function is moved left or right along the x-axis. To shift a function $f(x)$ to the right by $c$ units, we replace $x$ with $x - c$ in the function's equation. Conversely, to shift the function to the left by $c$ units, we replace $x$ with $x + c$.
Vertical Shifts
A vertical shift occurs when a function is moved up or down along the y-axis. To shift a function $f(x)$ up by $c$ units, we add $c$ to the function's equation. Conversely, to shift the function down by $c$ units, we subtract $c$ from the function's equation.
Combination of Horizontal and Vertical Shifts
When a function is shifted both horizontally and vertically, we need to combine the two shifts. For example, to shift a function $f(x)$ to the right by $c$ units and up by $d$ units, we replace $x$ with $x - c$ and add $d$ to the function's equation.
Stretching and Compressing Functions
Stretching and compressing a function involves changing its scale along the x-axis or y-axis. There are two types of stretches: horizontal and vertical.
Horizontal Stretching
A horizontal stretch occurs when a function is stretched along the x-axis. To stretch a function $f(x)$ by a factor of $k$, we replace $x$ with $x/k$ in the function's equation.
Vertical Stretching
A vertical stretch occurs when a function is stretched along the y-axis. To stretch a function $f(x)$ by a factor of $k$, we multiply the function's equation by $k$.
Reflecting Functions
Reflecting a function involves flipping its graph over a line or a point. There are two types of reflections: horizontal and vertical.
Horizontal Reflections
A horizontal reflection occurs when a function is flipped over the x-axis. To reflect a function $f(x)$ over the x-axis, we replace $f(x)$ with $-f(x)$ in the function's equation.
Vertical Reflections
A vertical reflection occurs when a function is flipped over the y-axis. To reflect a function $f(x)$ over the y-axis, we replace $x$ with $-x$ in the function's equation.
Applying Transformations to the Function $f(x) = x^2$
Now that we've discussed the different types of transformations, let's apply them to the function $f(x) = x^2$. We'll start by shifting the function to the right by 1 unit, vertically stretching it by a factor of 4, and shifting it down by 7 units.
Shifting the Function to the Right by 1 Unit
To shift the function $f(x) = x^2$ to the right by 1 unit, we replace $x$ with $x - 1$ in the function's equation:
Vertically Stretching the Function by a Factor of 4
To vertically stretch the function $g(x) = (x - 1)^2$ by a factor of 4, we multiply the function's equation by 4:
Shifting the Function Down by 7 Units
To shift the function $g(x) = 4(x - 1)^2$ down by 7 units, we subtract 7 from the function's equation:
The final function $g(x)$ is:
Conclusion
In this article, we've explored the process of shifting, stretching, and reflecting graphs of functions. We've applied these transformations to the function $f(x) = x^2$ to create a new function $g(x)$ that meets specific requirements. By understanding how to transform functions, we can create new functions that better model real-world phenomena or solve complex mathematical problems.
Example Problems
- Shift the function $f(x) = x^3$ to the left by 2 units and up by 3 units.
- Vertically stretch the function $g(x) = (x - 2)^2$ by a factor of 5.
- Reflect the function $h(x) = x^4$ over the x-axis and then shift it to the right by 1 unit.
Solutions
- To shift the function $f(x) = x^3$ to the left by 2 units and up by 3 units, we replace $x$ with $x + 2$ and add 3 to the function's equation:
- To vertically stretch the function $g(x) = (x - 2)^2$ by a factor of 5, we multiply the function's equation by 5:
- To reflect the function $h(x) = x^4$ over the x-axis, we replace $h(x)$ with $-h(x)$ in the function's equation:
Then, to shift the function to the right by 1 unit, we replace $x$ with $x - 1$ in the function's equation:
h(x) = -(x - 1)^4$<br/> # **Transforming Functions: A Q&A Guide** ## **Introduction** In our previous article, we explored the process of shifting, stretching, and reflecting graphs of functions. We applied these transformations to the function $f(x) = x^2$ to create a new function $g(x)$ that meets specific requirements. In this article, we'll answer some frequently asked questions about transforming functions. ## **Q&A** ### **Q: What is the difference between a horizontal shift and a vertical shift?** A: A horizontal shift occurs when a function is moved left or right along the x-axis, while a vertical shift occurs when a function is moved up or down along the y-axis. ### **Q: How do I shift a function to the right by c units?** A: To shift a function $f(x)$ to the right by $c$ units, you replace $x$ with $x - c$ in the function's equation. ### **Q: How do I shift a function up by d units?** A: To shift a function $f(x)$ up by $d$ units, you add $d$ to the function's equation. ### **Q: What is the difference between a horizontal stretch and a vertical stretch?** A: A horizontal stretch occurs when a function is stretched along the x-axis, while a vertical stretch occurs when a function is stretched along the y-axis. ### **Q: How do I stretch a function by a factor of k?** A: To stretch a function $f(x)$ by a factor of $k$, you replace $x$ with $x/k$ in the function's equation for a horizontal stretch, or you multiply the function's equation by $k$ for a vertical stretch. ### **Q: How do I reflect a function over the x-axis?** A: To reflect a function $f(x)$ over the x-axis, you replace $f(x)$ with $-f(x)$ in the function's equation. ### **Q: How do I reflect a function over the y-axis?** A: To reflect a function $f(x)$ over the y-axis, you replace $x$ with $-x$ in the function's equation. ### **Q: Can I apply multiple transformations to a function?** A: Yes, you can apply multiple transformations to a function. For example, you can shift a function to the right by $c$ units, then stretch it by a factor of $k$, and finally reflect it over the x-axis. ### **Q: How do I apply multiple transformations to a function?** A: To apply multiple transformations to a function, you need to follow the order of operations. For example, if you want to shift a function $f(x)$ to the right by $c$ units, then stretch it by a factor of $k$, and finally reflect it over the x-axis, you would replace $x$ with $x - c$, then replace $x$ with $x/k$, and finally replace $f(x)$ with $-f(x)$. ## **Example Problems** 1. Shift the function $f(x) = x^3$ to the left by 2 units and up by 3 units, then stretch it by a factor of 5. 2. Reflect the function $g(x) = (x - 2)^2$ over the x-axis, then shift it to the right by 1 unit and down by 7 units. 3. Apply multiple transformations to the function $h(x) = x^4$, including shifting it to the right by 3 units, stretching it by a factor of 2, and reflecting it over the y-axis. ## **Solutions** 1. To shift the function $f(x) = x^3$ to the left by 2 units and up by 3 units, then stretch it by a factor of 5, we replace $x$ with $x + 2$, add 3 to the function's equation, and then replace $x$ with $x/5$: $g(x) = 5(x + 2)^3 + 3
- To reflect the function $g(x) = (x - 2)^2$ over the x-axis, then shift it to the right by 1 unit and down by 7 units, we replace $g(x)$ with $-g(x)$, replace $x$ with $x - 1$, and then subtract 7 from the function's equation:
- To apply multiple transformations to the function $h(x) = x^4$, including shifting it to the right by 3 units, stretching it by a factor of 2, and reflecting it over the y-axis, we replace $x$ with $x - 3$, replace $x$ with $x/2$, and then replace $x$ with $-x$: