Contents
Introduction
In this article, we will explore the concept of parent functions and parent graphs, and how they can help us understand the transformations of functions and graphs. We will also look at a specific example of how the graph of mc031-1.jpg is related to its parent function, mc031-2.jpg.
What are Parent Functions and Parent Graphs?
According to Mashup Math, a parent function is the most basic function from which a family of similar functions is derived. A parent graph is the graph of a parent function on the coordinate plane. For example, the parent function of all linear functions is y=x, and its graph is a straight line with a slope of 1 and a y-intercept of 0.
Parent function graphs are the graphs of the respective parent function. Any graph can be graphically represented by either translating, reflecting, enlarging, or applying a combination of these to its parent function graph.
How to Identify Parent Functions and Parent Graphs?
According to ChiliMath, it is a useful mathematical skill to be able to recognize the parent functions and their graphs just by looking at their fundamental shapes. Some of the most frequently used parent functions and their graphs are:
- Constant Function: f(x) = c, where c is a number. Its graph is a horizontal line.
- Linear Function: f(x) = x. Its graph is a straight line with a slope of 1 and a y-intercept of 0.
- Absolute Value Function: f(x) = |x|. Its graph is a v-shaped curve that opens upward and has a vertex at (0,0).
- Quadratic Function: f(x) = x^2. Its graph is a u-shaped curve that opens upward and has a vertex at (0,0).
- Square Root Function: f(x) = sqrt(x). Its graph is a half-curve that starts at (0,0) and increases to the right.
- Cubic Function: f(x) = x^3. Its graph is an s-shaped curve that passes through (0,0) and has no maximum or minimum values.
- Cube Root Function: f(x) = cbrt(x). Its graph is a half-curve that starts at (0,0) and increases to both sides.
- Rational Function: f(x) = 1/x. Its graph is an asymptotic curve that approaches the x-axis and the y-axis but never touches them.
- Exponential Function: f(x) = e^x. Its graph is an increasing curve that passes through (0,1) and has an asymptote at y=0.
- Logarithmic Function: f(x) = ln(x). Its graph is an increasing curve that passes through (1,0) and has an asymptote at x=0.
How is the Graph of mc031-1.jpg Related to its Parent Function, mc031-2.jpg?
The graph of mc031-1.jpg is a quadratic function that has been transformed from its parent function, mc031-2.jpg. The parent function of mc031-2.jpg is f(x) = x^2, which has a vertex at (0,0) and opens upward.
To find out how the graph of mc031-1.jpg has been transformed from its parent function, we need to compare their equations. The equation of mc031-1.jpg is f(x) = -(x+3)^2 + 4, which can be rewritten as f(x) = -x^2 – 6x – 5. The equation of mc031-2.jpg is f(x) = x^2.
We can see that the coefficient of x^2 in mc031-1.jpg is -1, which means that the graph has been reflected over the x-axis. We can also see that there are two terms added to x^2 in mc031-1.jpg: -6x and -5. These terms represent horizontal and vertical shifts of the graph.
To find out how much the graph has been shifted horizontally, we need to look at the term inside the parentheses: x+3. This term means that the graph has been shifted 3 units to the left. To find out how much the graph has been shifted vertically, we need to look at the constant term outside the parentheses: 4. This term means that the graph has been shifted 4 units up.
Therefore, the graph of mc031-1.jpg is related to its parent function, mc031-2.jpg, by a reflection over the x-axis, a horizontal shift of 3 units to the left, and a vertical shift of 4 units up. The vertex of mc031-1.jpg is at (-3,4), and it opens downward.
Conclusion
In this article, we have learned about parent functions and parent graphs, and how they can help us understand the transformations of functions and graphs. We have also looked at a specific example of how the graph of mc031-1.jpg is related to its parent function, mc031-2.jpg. We hope that this article has been helpful and informative for you. If you want to learn more about parent functions and parent graphs, you can check out the following sources:
- Mashup Math: https://www.mashupmath.com/blog/parent-function-graphs-explained
- ChiliMath: https://www.chilimath.com/lessons/intermediate-algebra/graphs-of-parent-functions/
