In math, the possibilities of making equations remain endless.

Yet, most or all equations are a result of transformations and eventually tone down to a parent functionTo learn how to find parent functions through equations and/or graphs, follow the methods below:

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How to Find Parent Functions through Equations

Step 1:

Observe the question.

Whether you are finishing up on a math assignment, going through a quiz, or conducting a class – the first step to finding the parent function is to examine the equation in question. Have the equation written neatly in front of you. Most equations have “y” or “f(x)” as a subject. They are then followed be an “equal to” sign and contain variables and constants.

Observe the inputs of an equation, the subject, and other variables. Know what each of them stand for.

Step 2:

Search for signs and eliminate them.

When we say search for signs, we certainly don’t mean look for the heavens or the earth to grant you a sign. In maths, signs are mathematical operation like a plus, minus, divide, or multiply. These determine the slope and trends of equations and whether they are increasing, decreasing, etc.

With parent functions, you don’t need such signs. Therefore, eliminate such mathematical operators from your equation to simplify it. For example, when trying how to find parent functions of the equation:

Strip the minus sign off the equation and you shall arrive at the parent function. That is, ; a line passing through the origin with gradient 1.

Step 3:

Look for constants and remove them.

For those unaware, constants result in horizontal and vertical transformations in an equation. They are not within the parent function. Therefore, when deriving the parent function from a given equation, simply delete or erase the constants.

If you are worried you may delete something important from an equation; there is a tip to figuring out the constant. Look for numbers unaccompanied by variables. These can be mere additions or deductions within a function.

For example:

In this equation, we first eliminate the sign – resulting in. This seems a bit weird to be the parent function. Therefore, we look for the constant; ‘2’ in this case and remove it.

The simplified parent function is:

Step 4:

Next, remove coefficients of variables.

Variable coefficients are the numbers accompanying variables in an equation. They have a direct impact on the gradient, and thus determine the pace of growth/fall of the equation. However, they aren’t of much use in the parent function. Therefore, to learn how to find parent functions, you should remove the coefficients.

For example:

This equation has no sign or constant. However, it has a variable coefficient: 5.

To arrive upon your parent function, delete the coefficient, resulting in .

Step 5:

Make power adjustments.

Contrary to common thought, these adjustments do make their way to powers, instead of remaining confined to the main equation. Thus, observe a function for signs, constants, and variable coefficients and make these adjustments in the power too.

For example, most would simplify  to  . However, this is incorrect. The right way to learn how to find the parent function is to make changes in the power accordingly. Therefore, the right simplification to a parent function will be:

Step 6:

Write down the final function.

Congratulations, you have learnt how to find parent functions.

How to Find Parent Functions through Graphs

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Step 1:

Know the shape of parent graphs.

Different shapes indicate different parent functions. It is important to know the general shape of parent graphs before embarking on the mission of how to find parent functions through graphs.

There are a few types of parent functions:

  1. Constant
  2. Linear
  • Absolute Value
  1. Quadratic
  2. Cubic
  3. Logarithmic
  • Exponential
  • Cube Root
  1. Square Root
  2. Trigonometric Graphs (e.g. those of sin x, cos x, tan x, etc.)

All other graphs are generally derived from the shapes of these.

Step 2:

Observe the shape of the function in question.

After you know the general shape of parent functions, the rest of the answer essentially lies in matching the given question with the right parent.

By simply observing the shape, you can get a rough idea of what the parent function should be. For example: while a simple line can be indicative of a linear equation, a parabola can hint at a quadratic equation. A wave-like function symbolizes graphs of sin (x) and so on.

Step 3:

Adjust for sign, constant, and coefficient variable.

As is the case with general equations, you may be confused in matching the graph to the right parent. This is because graphs can appear similar yet not the same. Do not fret, this is exactly how it should be. The graphs are never fully accurate to the parent function and requires adjustments:

  • Eliminate the effects of sign. For example, regardless of whether a line is negatively-sloped, it shall be considered positive (and its sign negated) for the parent function.
  • Disregard y-intercepts. These are constants in the graph and play no role for the parent function. For example, a y-intercept of y=x+5 only indicates shifting up 5 units of the parent graph; that is y=x in this case.
  • Eradicate coefficient variables as they simply stretch or squeeze a graph. For example, a graph of has a parent function . You can see in the graph, that the 2 only squeezes the parabola.

Step 4:

Write your parent function down, and you have learnt how to find parent functions.

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