Graphing Functions: Absolute Value, Cubic, Quadratic

Hey guys! Let's dive into graphing some functions. We're going to sketch the graphs of a few different types of functions, including absolute value, cubic, and quadratic functions. We'll also explore how transformations affect these graphs. So, grab your graph paper (or your favorite graphing software) and let's get started!

1. Sketching the Graph of $f(x) = 2|x+3| - 1$

Alright, let's tackle the absolute value function first. Our mission is to sketch the graph of f(x) = 2|x + 3| - 1. To nail this, we need to understand how the different parts of this equation transform the basic absolute value function, |x|. Think of it like this: we're taking the simple |x| graph and giving it a bit of a makeover.

First, let's consider the |x + 3| part. This shifts the basic absolute value graph 3 units to the left. Remember, it's the opposite of what you might think! So, instead of moving to the right, we're sliding the whole graph over to the left. This is a horizontal translation, and it's a key part of understanding how the function behaves.

Next up, we have the 2|x + 3|. Multiplying the absolute value by 2 stretches the graph vertically by a factor of 2. This means the graph becomes narrower, and the V-shape becomes more pronounced. Every y-value is doubled, making the graph steeper. This is a vertical stretch, and it changes the overall shape of the graph.

Finally, we have the -1 at the end. This shifts the entire graph down by 1 unit. So, after shifting left and stretching vertically, we bring the whole thing down a notch. This is a vertical translation, and it completes the transformation of the basic absolute value function.

To sketch the graph, start by plotting the vertex. The vertex of the basic |x| function is at (0, 0). After the horizontal shift of 3 units to the left, the vertex moves to (-3, 0). Then, after the vertical shift of 1 unit down, the vertex ends up at (-3, -1). So, that's our starting point.

Now, consider the vertical stretch. The slope of the right side of the basic |x| function is 1. After the vertical stretch by a factor of 2, the slope becomes 2. This means for every 1 unit you move to the right, you move 2 units up. Similarly, the slope of the left side is -2.

So, from the vertex (-3, -1), sketch a line with a slope of 2 to the right and a line with a slope of -2 to the left. These two lines form the V-shape of the absolute value function. And there you have it! The graph of f(x) = 2|x + 3| - 1.

In summary, to graph f(x) = 2|x + 3| - 1, you:

1. Shift the basic absolute value function |x| 3 units to the left.
2. Stretch the graph vertically by a factor of 2.
3. Shift the graph 1 unit down.

By understanding these transformations, you can easily sketch the graph of any absolute value function in this form. Remember to pay attention to the order of the transformations and how they affect the position and shape of the graph.

2. Sketching the Graph of $y = \frac{1}{2}x^3$

Now, let's move on to the cubic function. We want to sketch the graph of y = (1/2)x³. This is a transformation of the basic cubic function, y = x³. The key here is to understand how the coefficient 1/2 affects the shape of the graph.

The basic cubic function, y = x³, passes through the origin (0, 0) and has a characteristic S-shape. It increases slowly for negative values of x, then increases rapidly for positive values of x. The graph is symmetric about the origin, meaning it has rotational symmetry of 180 degrees.

Now, let's consider the (1/2) in front of the x³. This is a vertical compression (or stretch) by a factor of 1/2. It means that every y-value of the basic cubic function is multiplied by 1/2. This makes the graph wider and less steep than the basic cubic function.

To sketch the graph, start by plotting a few key points. When x = 0, y = (1/2)(0)³ = 0, so the graph still passes through the origin. When x = 1, y = (1/2)(1)³ = 1/2. When x = 2, y = (1/2)(2)³ = 4. When x = -1, y = (1/2)(-1)³ = -1/2. And when x = -2, y = (1/2)(-2)³ = -4.

Plot these points and then connect them with a smooth curve. The graph will still have the S-shape characteristic of a cubic function, but it will be wider and less steep than the graph of y = x³. The vertical compression makes the graph appear to be squashed down towards the x-axis.

To summarize, to graph y = (1/2)x³, you:

1. Start with the basic cubic function y = x³.
2. Compress the graph vertically by a factor of 1/2.

This vertical compression makes the graph wider and less steep. Remember, a coefficient between 0 and 1 will always compress the graph vertically, while a coefficient greater than 1 will stretch the graph vertically.

By understanding how coefficients affect the shape of the graph, you can easily sketch the graph of any cubic function in this form. Just remember to pay attention to the value of the coefficient and how it affects the y-values of the function.

3. Sketching the Graph of $y = -2x^2 + 3$

Next up, let's sketch the graph of the quadratic function y = -2x² + 3. This is a transformation of the basic quadratic function, y = x². We need to understand how the -2 and the +3 affect the shape and position of the graph.

The basic quadratic function, y = x², is a parabola that opens upwards and has its vertex at the origin (0, 0). It's a symmetrical U-shape that increases rapidly as you move away from the origin in either direction.

Now, let's consider the -2 in front of the x². The negative sign reflects the graph across the x-axis, so instead of opening upwards, the parabola opens downwards. The 2 stretches the graph vertically by a factor of 2, making it narrower than the basic parabola.

Finally, the +3 shifts the entire graph upwards by 3 units. This moves the vertex from the origin (0, 0) to the point (0, 3). So, the parabola is now upside down and shifted upwards.

To sketch the graph, start by plotting the vertex at (0, 3). Since the parabola opens downwards, the graph will go down from this point. The vertical stretch of 2 means that for every 1 unit you move to the right or left, you move 2 units down.

So, plot the points (1, 1) and (-1, 1). These points are 1 unit to the right and left of the vertex, and 2 units down from the vertex. Then, connect these points with a smooth curve to form the parabola. The graph will be upside down and narrower than the basic parabola.

To summarize, to graph y = -2x² + 3, you:

1. Reflect the basic parabola y = x² across the x-axis.
2. Stretch the graph vertically by a factor of 2.
3. Shift the graph 3 units up.

This combination of transformations creates a parabola that opens downwards, is narrower than the basic parabola, and has its vertex at (0, 3). Remember to pay attention to the order of the transformations and how they affect the shape and position of the graph.

4. Sketching the Graph of $y = (x+2)^2 - 1$

Okay, let's sketch the graph of y = (x + 2)² - 1. This is another quadratic function, but this time it's in vertex form. This form makes it easy to identify the vertex of the parabola and how it's been shifted from the basic quadratic function, y = x².

As we mentioned before, the basic quadratic function, y = x², is a parabola that opens upwards and has its vertex at the origin (0, 0). It's symmetrical about the y-axis and increases rapidly as you move away from the origin.

In the equation y = (x + 2)² - 1, the (x + 2)² part shifts the graph horizontally. Remember, it's the opposite of what you might think! So, the +2 shifts the graph 2 units to the left. This moves the vertex from (0, 0) to (-2, 0).

The -1 at the end shifts the entire graph vertically. In this case, it shifts the graph 1 unit down. So, after shifting left, we bring the whole thing down a notch. This moves the vertex from (-2, 0) to (-2, -1).

To sketch the graph, start by plotting the vertex at (-2, -1). Since the coefficient of the (x + 2)² term is positive (it's 1), the parabola opens upwards. The shape of the parabola is the same as the basic parabola, y = x², since there's no vertical stretch or compression.

So, from the vertex (-2, -1), sketch a parabola that opens upwards. The points 1 unit to the right and left of the vertex will be 1 unit up from the vertex. So, plot the points (-1, 0) and (-3, 0). These points are on the parabola.

Connect these points with a smooth curve to form the parabola. The graph will be the same shape as the basic parabola, but it will be shifted 2 units to the left and 1 unit down. And there you have it! The graph of y = (x + 2)² - 1.

To summarize, to graph y = (x + 2)² - 1, you:

1. Shift the basic parabola y = x² 2 units to the left.
2. Shift the graph 1 unit down.

These transformations create a parabola that opens upwards and has its vertex at (-2, -1). Remember, the vertex form of a quadratic function, y = a(x - h)² + k, makes it easy to identify the vertex (h, k) and how the graph has been shifted from the basic parabola.

5. Understanding the Relationship Between $f(x)$ and $f(3x)$

Finally, let's talk about the relationship between f(x) and f(3x). This involves a horizontal compression of the graph. When you replace x with 3x in a function, you're essentially squeezing the graph horizontally towards the y-axis.

To understand this, consider what happens to the x-values. For any given y-value, the x-value in f(3x) is one-third of the x-value in f(x). This is because to get the same y-value, you need to plug in a smaller x-value into f(3x).

For example, if f(2) = 5, then f(3 * (2/3)) = f(2) = 5. So, f(3x) = 5 when x = 2/3. This means the point (2, 5) on the graph of f(x) corresponds to the point (2/3, 5) on the graph of f(3x). The x-value has been compressed by a factor of 3.

In general, the graph of f(3x) is the graph of f(x) compressed horizontally by a factor of 3. This means the graph becomes narrower and closer to the y-axis. The y-values remain the same, but the x-values are squeezed together.

To visualize this, imagine taking the graph of f(x) and squeezing it horizontally like an accordion. The y-axis stays fixed, and the graph is pushed towards the y-axis. The amount of compression depends on the factor multiplying x. In this case, the factor is 3, so the graph is compressed by a factor of 3.

So, the key takeaway here is that replacing x with 3x in a function compresses the graph horizontally by a factor of 3. This is a useful transformation to understand, as it can help you quickly sketch the graph of a transformed function without having to plot a lot of points.

And that's a wrap, guys! We've covered how to sketch the graphs of absolute value, cubic, and quadratic functions, and we've also explored how transformations affect these graphs. Remember to practice these techniques, and you'll become a graphing pro in no time!