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# Manipulating quadratic and exponential expressions | Lesson

## What are "manipulating quadratic and exponential expressions" questions, and how frequently do they appear on the test?

Note: On your official SAT, you might not see a question on manipulating quadratic and exponential expressions at all! At most, you'll see 1 question.
If you haven’t already mastered more frequently tested SAT skills, you may want to save this topic for later.

Manipulating quadratic and exponential expressions questions can ask us to rewrite an expression to showcase a specific graphical feature. For example, given the equation y, equals, x, squared, plus, 3, x, minus, 4, we may be asked to rewrite x, squared, plus, 3, x, minus, 4 in a way that shows the x-intercepts of the graph.
We may also be asked to write equivalent expressions based on real-world scenarios. An especially challenging question may ask us to change the time unit of exponential expressions, e.g., changing a monthly population growth model to an equivalent annual population growth model.
This lesson builds upon the following skills:
You can learn anything. Let's do this!

## How do I rewrite quadratic expressions to reveal specific features of parabolas?

### Forms & features of quadratic functions

Forms & features of quadratic functionsSee video transcript

### Equivalent forms of quadratic expressions

The Graphing quadratic functions lesson covers the three forms of quadratic functions and the graphical features they display as constants and coefficients.
When we're given a quadratic function, we can rewrite the function according to the features we want to display:
• y-intercept: standard form
• x-intercept(s): factored form
• Vertex: vertex form
When rewriting a quadratic function to display specific graphical features:
1. Choose the appropriate quadratic form based on the graphic feature to be displayed.
2. Rewrite the given quadratic expression as an equivalent expression in the form identified in Step 1.

Example:
The graph of y, equals, x, squared, minus, 4, x, minus, 5 is shown above. Write an equivalent equation from which the coordinates of the vertex can be identified as constants in the equation.

### Try it!

Try: identify the appropriate forms to use
A graph of the quadratic equation y, equals, x, squared, minus, 2, x, minus, 3 is shown in the x, y-plane above.
The vertex of the parabola is located at
. To show the coordinates of the vertex as constants or coefficients, we should use the
form of the quadratic expression, y, equals, left parenthesis, x, minus, 1, right parenthesis, squared, minus, 4.
The x-intercepts of the parabola are
. To show the x-intercepts as constants or coefficients, we should use the
form of the quadratic expression, y, equals, left parenthesis, x, plus, 1, right parenthesis, left parenthesis, x, minus, 3, right parenthesis.

## How do I rewrite exponential expressions using different time units?

### Interpreting time in exponential models

Interpreting time in exponential modelsSee video transcript

### How do we change time units in an exponential expression?

On the SAT, we're sometimes given an exponential model in one time unit (for example: years), and asked to re-write it using another time unit (for example: months), showing the same rate of growth or decay as the original model.
For example, let's look at the equation M, equals, 100, left parenthesis, 1, point, 62, right parenthesis, start superscript, t, end superscript, where M is the number of members a club has t years after it opens. How do we write an equation for the number of club members m months after opening?
In this type of question, the base of the exponent is kept the same, and only the exponent changes. We know that the number of club members increases by a factor of 1, point, 62 after t, equals, 1 year, which means the number of club members also needs to increase by a factor of 1, point, 62 after m, equals, 12 months.
To make sure the exponent of 1, point, 62 is 1 when m, equals, 12 and 2 when m, equals, 24 (two years), we need to divide m by 12 in the exponent:
M, equals, 100, left parenthesis, 1, point, 62, right parenthesis, start superscript, start superscript, start fraction, m, divided by, 12, end fraction, end superscript, end superscript
When changing the time units in an exponential expression:
• Identify the unit equivalence used for the question, e.g., 1, start text, y, e, a, r, end text, equals, 12, start text, m, o, n, t, h, s, end text.
• Tip: If converting the variable from a larger time unit to a smaller time unit, we need to make the exponent smaller. For example, when converting from years to months, we divide the variable representing the number of months by 12.
• Tip: If converting the variable from a smaller time unit to a larger time unit, we need to make the exponent larger. For example, when converting from months to years, we multiply the variable representing the number of years by 12.
• Verify that the two expressions give us the same values when evaluated using equivalent amounts of time, e.g., 1 year and 12 months, 2 years and 24 months, etc.

### Try it!

Try: convert from hours to days
P, equals, 50, left parenthesis, 1, point, 1, right parenthesis, start superscript, h, end superscript
The equation above models the population, P, of a bacteria culture after h hours of incubation.
There are 24 hours in a day. Since a day is
than an hour, if we want to write an equation that models the population of the bacteria culture after d days of incubation, we must
the exponent by 24.
Which of the following equations models the population of the bacteria culture after after d days of incubation?

Practice: identify the graphical feature displayed as a constant or coefficient
If y, equals, left parenthesis, x, minus, 2, right parenthesis, squared, plus, 6 is graphed in the x, y-plane. which of the following characteristics of the graph is displayed as a constant or coefficient in the equation?

Practice: change the time unit in an exponential model
D, equals, 200, left parenthesis, 1, point, 16, right parenthesis, start superscript, m, end superscript
The equation above models the number of total downloads, D, for an app Clara created m months after its launch. Of the following, which equation models the number of total downloads y years after launch?