Graph Shift: Finding The New Equation

Hey guys! Let's dive into a cool problem about graph transformations. We're going to figure out what happens to the equation of a graph when we shift it up or down. Specifically, we'll look at the function $f(x) = 2^x$ and see what its new equation is when we move it 3 units up. Understanding these transformations is super helpful in math, especially when you're dealing with functions and graphs. So, let's break it down step by step!

Understanding the Original Function: $f(x) = 2^x$

Before we shift anything, let's get comfy with the original function, $f(x) = 2^x$. This is an exponential function, which means that as $x$ increases, $f(x)$ increases really, really fast. The graph starts very close to the x-axis on the left side (when x is a big negative number) and shoots up dramatically on the right side (when x is a big positive number).

Key characteristics of $f(x) = 2^x$ include:

• It always passes through the point (0, 1) because $2^0 = 1$.
• It gets closer and closer to the x-axis as $x$ goes towards negative infinity, but it never actually touches the x-axis. This is called an asymptote.
• As $x$ increases, $f(x)$ increases exponentially, meaning it grows at an increasing rate.

Knowing these characteristics helps us understand how the graph will behave when we start moving it around. Now, let's see what happens when we shift it upwards!

Shifting the Graph Upwards

Okay, so we want to shift the graph of $f(x) = 2^x$ upwards by 3 units. What does that mean for the equation? Well, when you shift a graph up or down, you're basically changing the y-coordinate of every point on the graph. If we're moving the graph up by 3 units, we're adding 3 to every y-coordinate.

So, if a point on the original graph is $(x, y)$, the corresponding point on the shifted graph will be $(x, y + 3)$. This means that if the original function gives us $y = 2^x$, the new function will give us $y + 3 = 2^x$.

To find the equation of the new graph, we need to solve for $y$ in terms of $x$. So, we subtract 3 from both sides of the equation $y + 3 = 2^x$, which gives us $y = 2^x + 3$.

The New Equation: $y = 2^x + 3$

So, the equation of the new graph after shifting $f(x) = 2^x$ upwards by 3 units is $y = 2^x + 3$. This is the equation that represents the transformed graph. Let's think about why this makes sense:

• For any value of $x$, the new $y$ value is always 3 more than the original $y$ value.
• The graph still has the same basic shape as $f(x) = 2^x$, but it's been lifted up by 3 units.
• The new graph passes through the point (0, 4) because when $x = 0$, $y = 2^0 + 3 = 1 + 3 = 4$.

Therefore, the correct answer is B. $y = 2^x + 3$.

Why the Other Options Are Wrong

It's always a good idea to understand why the other options are incorrect. Let's take a quick look:

• A. $y = 2^x - 3$: This would shift the graph down by 3 units, not up.
• C. $y = 2^{x-3}$: This would shift the graph to the right by 3 units. Remember, changes inside the function (affecting $x$) cause horizontal shifts, and a minus sign shifts it to the right.
• D. $y = 2^{x+3}$: This would shift the graph to the left by 3 units. Again, changes inside the function cause horizontal shifts, and a plus sign shifts it to the left.
• E. $y = 3 eq 2^x$: This would stretch the graph vertically by a factor of 3. It's a vertical stretch, not a vertical shift.

Understanding why these options are wrong helps solidify your understanding of graph transformations.

General Rules for Graph Transformations

To wrap things up, let's look at some general rules for graph transformations. Knowing these rules will help you tackle all sorts of graph-shifting problems.

Vertical Shifts

• To shift a graph up by $k$ units, add $k$ to the function: $y = f(x) + k$.
• To shift a graph down by $k$ units, subtract $k$ from the function: $y = f(x) - k$.

Horizontal Shifts

• To shift a graph to the right by $h$ units, replace $x$ with $(x - h)$ in the function: $y = f(x - h)$.
• To shift a graph to the left by $h$ units, replace $x$ with $(x + h)$ in the function: $y = f(x + h)$.

Vertical Stretches and Compressions

• To stretch a graph vertically by a factor of $a$, multiply the function by $a$: $y = a eq f(x)$. If $a > 1$, it's a stretch; if $0 < a < 1$, it's a compression.

Horizontal Stretches and Compressions

• To stretch a graph horizontally by a factor of $b$, replace $x$ with $(x/b)$ in the function: $y = f(x/b)$. If $b > 1$, it's a stretch; if $0 < b < 1$, it's a compression.

Reflections

• To reflect a graph across the x-axis, multiply the function by -1: $y = -f(x)$.
• To reflect a graph across the y-axis, replace $x$ with $-x$ in the function: $y = f(-x)$.

Understanding these rules makes it much easier to visualize and manipulate graphs. Remember, practice makes perfect, so try applying these rules to different functions and see how they transform!

Example Problems

Let's try a couple more quick examples to make sure we've got this down.

Example 1:

The graph of $y = x^2$ is shifted 2 units to the left. What is the new equation?

Solution: Shifting to the left means we replace $x$ with $(x + 2)$. So, the new equation is $y = (x + 2)^2$.

Example 2:

The graph of $y = |x|$ is reflected across the x-axis and then shifted up by 1 unit. What is the new equation?

Solution: Reflecting across the x-axis means we multiply the function by -1, giving us $y = -|x|$. Then, shifting up by 1 unit means we add 1 to the function, giving us $y = -|x| + 1$.

By working through these examples, you can see how the transformation rules apply in different situations. Keep practicing, and you'll become a graph transformation pro in no time!

Conclusion

Alright, guys, that wraps up our exploration of graph shifts! We've seen how shifting the graph of $f(x) = 2^x$ upwards by 3 units changes its equation to $y = 2^x + 3$. We also covered the general rules for vertical and horizontal shifts, stretches, compressions, and reflections. Armed with this knowledge, you'll be able to tackle a wide range of graph transformation problems.

Remember, the key to mastering these concepts is practice. So, keep graphing, keep shifting, and keep exploring! You've got this!