In this article, we will explore how the graph of a function can be transformed by changing its equation. We will use the example of mc027-1.jpg and mc027-2.jpg to illustrate the concepts of parent functions and transformations.
Contents
What is a parent function?
A parent function is a basic function that can be used as a starting point to create other functions by applying transformations. Some common parent functions are:
– Linear: $$y=x$$
– Quadratic: $$y=x^2$$
– Absolute value: $$y=|x|$$
– Square root: $$y=\sqrt{x}$$
– Cubic: $$y=x^3$$
– Exponential: $$y=b^x$$
– Logarithmic: $$y=\log_b x$$
The graph of a parent function has a characteristic shape that can be recognized easily. For example, the graph of the linear function is a straight line, the graph of the quadratic function is a parabola, and the graph of the absolute value function is a V-shaped curve.
What are function transformations?
Function transformations are changes in the equation of a function that affect its graph. There are four main types of transformations:
– Vertical shift: This moves the graph up or down by adding or subtracting a constant to the function. For example, $$y=x+3$$ is a vertical shift of 3 units up from $$y=x$$.
– Horizontal shift: This moves the graph left or right by adding or subtracting a constant to the input variable. For example, $$y=(x-2)^2$$ is a horizontal shift of 2 units right from $$y=x^2$$.
– Vertical stretch or compression: This changes the height of the graph by multiplying or dividing the function by a constant. For example, $$y=2x^2$$ is a vertical stretch of factor 2 from $$y=x^2$$.
– Horizontal stretch or compression: This changes the width of the graph by multiplying or dividing the input variable by a constant. For example, $$y=\sqrt{2x}$$ is a horizontal compression of factor 0.5 from $$y=\sqrt{x}$$.
There are also other types of transformations, such as reflection, rotation, and dilation, but we will focus on the four main ones in this article.
How to relate the graph of mc027-1.jpg to its parent function, mc027-2.jpg?
To relate the graph of mc027-1.jpg to its parent function, mc027-2.jpg, we need to identify the equation of each function and compare them. We can see that mc027-2.jpg is the graph of the parent function $$y=\sqrt{x}$$, which has a domain of $$[0,\infty)$$ and a range of $$[0,\infty)$$.
The equation of mc027-1.jpg is not given, but we can find it by using some clues from the graph. We can see that:
– The graph has been shifted 3 units left and 4 units down from the parent function. This means that we need to add 3 to the input variable and subtract 4 from the function.
– The graph has been stretched vertically by a factor of 2 from the parent function. This means that we need to multiply the function by 2.
Therefore, the equation of mc027-1.jpg is:
$$y=2\sqrt{x+3}-4$$
We can verify this by plugging in some values from the graph and checking if they satisfy the equation. For example, when $$x=-3$$, we have:
$$y=2\sqrt{-3+3}-4=2\sqrt{0}-4=-4$$
This matches with the point $(-3,-4)$ on the graph.
We can also use this equation to find the domain and range of mc027-1.jpg. Since we have a square root in the equation, we need to make sure that the expression inside it is non-negative. This means that:
$$x+3\geq 0 \implies x\geq -3$$
So, the domain of mc027-1.jpg is $$[-3,\infty)$$. To find the range, we need to consider the minimum and maximum values of y. The minimum value occurs when x is at its lowest value in the domain, which is -3. Plugging this into the equation, we get:
$$y_{min}=2\sqrt{-3+3}-4=-4$$
The maximum value occurs when x approaches infinity. As x gets larger and larger, y also gets larger and larger without bound. So, we can write:
$$y_{max}=\infty$$
Therefore, the range of mc027-1.jpg is $$[-4,\infty)$$.
Conclusion
In this article, we learned how to relate the graph of mc027-1.jpg to its parent function, mc027-2.jpg, by identifying the equation, domain, and range of each function and comparing them. We also learned how to apply the four main types of function transformations: vertical shift, horizontal shift, vertical stretch or compression, and horizontal stretch or compression. These transformations can help us create new functions from existing ones and understand how their graphs change accordingly.
