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### Course: Algebra 2>Unit 9

Lesson 7: Graphs of exponential functions

# Transforming exponential graphs (example 2)

Given the graph of y=2ˣ, Sal graphs y=(-1)2ˣ⁺³+4, which is a vertical reflection and a shift of y=2ˣ.

## Want to join the conversation?

• what is the difference between Geometric sequence equation and Exponential equation.
• Although both multiply by a common ratio to get to the next number, there are still some subtle differences between them. Geometric sequences are exactly that: sequences. They are a never-ending set of numbers in a list. An exponential equation can only be represented by a graph, and you can't put those values in a list (there are just too many).
• Shall I first learn graphs before getting into this story?
• Yes, start with how to graph points, then graphing linear functions, then graphing quadratic functions. Once you understand those, you should be okay for exponential graphs.
• Is the constant always going to be the horizontal asymptote?
• No, not always.
Consider the rational function f(x) = (ax^n + bx + 6) / (cx^m + dx + 6).
The horizontal asymptote is the horizontal line that f(x) approaches as x approaches positive/negative infinity. You don't have enough information to find that, but you do have enough information to find the y-intercept. What is f(x) when x is zero?

f(0) = 6/6 = 1
y-intercept: (0, 1)

Constants will not be an HA if you're also working with linear equations.
I'm not sure how else constants will be the HA, but it would be a good topic to research.
• In the first graph that Sal made, `y=2^x+3`, at . Why is the y intercept 5? I thought that if x is 0 then 2 would be to the power of 3 which is 8, right? or is it a error?
• As drawn by Sal, the y-intercept is 6. However, you are correct that it should be 8. The difference comes because Sal's drawing is only an approximate sketch. Further on in the video at about , there is a pop-up note that also mentions another inaccuracy of Sal's sketch.
• when Sal says y=2 to the power of x+3, i got confused. How did he get the power of x+3 simplified? What steps did he use?
• Sal had the graph y = 2^x.
The second graph, y = -1 * 2^(x+3) + 4, is a transformation of y = 2^x.
According to March11, Sal must use PEMDAS to track how y = 2^x will transform.
He first works with the exponent (E).
A 3 was added to the x in y = 2^x, making it y = 2^(x+3). Sal transforms the graph accordingly. Afterwards he moves on to the other parts of the transformation (multiplying -1 and adding 4).
There is no simplification involved, only working by chunks.
• Can someone tell me how to use this order of operation, because I am lost on where to start first?
• In school, we always learned PLEMDAS. Parentheses, Logarithms and Exponents, Multiplication and Division, Addition and Subtraction.
Within each "section" (like addition and subtraction) you do operations right to left. Take
3-2+1.
First, you subtract 2 from 3 and then add 1.

You do parentheses first because that's mathematical shorthand for "do these first!".

Next are exponentiation, square roots, and logarithms. Exponents are repeated multiplication, so we take care of them before moving on. If we have
2 * 2^3
We do the exponent before the multiplication. 2^3 is 8, times 2 is 16. Logarithms and roots also involve exponents, so they're in this "section" too.

Multiplication and division are next. Take
4/2 * x + 3
4 divided by 2 is 2, times x is 2x, plus 3 is 2x +3.

In Sal's problem,
-1 * 2^x+3 + 4
You change the graph according to the exponent first, then the multiplication, then the addition.

Essentially, You're moving from most complex to least complex. I'm a little late. Hope that helps. Mathematicians have so many rules XD
• Does anyone know of a faster way to find x values that work to plot? I keep going through all of the x values and solving for them, which is extremely time consuming and annoying if I solve for the correct plot-able x value incorrectly and have to start all over again.
• Can someone help me describe the transformations of y= -2(3^-x)+1
• The parent graph is y = 3^x
The transformations in order: reflection in y-axis, followed by reflection in x-axis, followed by vertical stretch by a factor 2, and finally a vertical shift upwards of 1 unit.