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A short class with Sarah and her students. The other ways are under More.
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Today, we're going to study how stretches or compressions affect cubic functions. Can anyone tell me what happens when we increase the value of 'a' in the function f(x) = a * x³?
I think it makes the graph taller?
Exactly! If |a| > 1, the graph stretches vertically, and if 0 < |a| < 1, we compress it. So, we can remember this with the saying: 'Tall a, stretch all; small a, compress the ball!'
What if a is negative? Does that change the direction as well?
Good question! A negative 'a' reflects the graph over the x-axis. So, keep in mind that the sign of 'a' plays a crucial role in the graph's appearance.
To sum it up, vertical stretching makes graphs taller and compressing flattens them, while negative values reflect them. Let’s remember our rules!
Now, let's discuss horizontal and vertical shifts. If we have f(x) = f(x - h) + k, how would the function change if we let h be 2 and k be 3?
That means we’d move the graph 2 units to the right and 3 units up, right?
Correct! This is a practical transformation; it helps us move our function around on the graph. Can someone give me a real-world example of where this might be useful?
If we were modeling something that experiences a consistent delay or elevation change, like a car on a hill?
Exactly! The ability to shift graphs allows us to adjust our models to better fit real-life contexts. Let’s summarize our findings: horizontal shifts change x, vertical shifts change y.
We can now combine what we’ve learned. If we take f(x) = 2(x - 1)³ + 3, what transformations are happening to the parent function x³?
It’s stretched vertically because of the 2, moved right by 1, and shifted up by 3.
Right, you all are getting better at this! How would you visualize this in steps?
Start at the basic graph of x³, stretch it taller, shift right, and then raise it higher.
Fantastic! Remember this step-by-step method to systematically apply transformations to any cubic function. Reviewing this will help you remember how to create a graph accurately.
Movements and changes applied to the cubic graph.
A change that makes the graph taller.
A change that makes the graph flatter.
A lateral movement of the graph left or right.
A move of the graph up or down.
Stretched up high, the graph will grow, compressed it’s flat, now you know!
Imagine a tree, tall and proud, but some days it gets pruned down loud. That’s how transformations make functions appear, taller or shorter, let’s give a cheer!
SHift is for shifting (h and k), S is for Stretching (vertically), and C is for Compression (when a < 1).
SVC: Stretch, Vertical, Compression.
Term
Vertical Stretch
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Term
Horizontal Shift
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Term
Vertical Compression
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Transformations
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A polynomial function of degree 3, generally expressed as f(x) = ax³ + bx² + cx + d.
A transformation that makes a graph taller.
A transformation that makes a graph flatter.
A transformation that moves the graph to a different location in the coordinate plane.
This section covers transformations applied to cubic functions, including vertical and horizontal shifts, and stretches or compressions.
Understanding transformations of cubic functions involves analyzing how changes in the function's equation affect the graph. This includes vertical and horizontal shifts, as well as stretching or compressing the graph. Key examples showcased transformations visually to aid understanding.
In this section, we explore how different transformations can alter the graph of cubic functions, which are polynomial functions of degree 3. The general form of a cubic function is given by
where a is not equal to zero. We will delve into two main types of transformations: vertical and horizontal shifts, and also vertical stretches or compressions.
A transformation can stretch or compress the cubic graph. This is denoted by multiplying the function by a factor a, resulting in:
If |a| > 1, there is a vertical stretch; if 0 < |a| < 1, the graph undergoes a vertical compression.
Cubic functions can also be translated or shifted. The transformation:
indicates a shift right by h units and up by k units. For example, consider the transformation:
This function is stretched vertically by a factor of 2, moved right by 1 unit, and shifted upward by 3 units.
Understanding these transformations is crucial for graphing cubic functions and interpreting their variations, making cubic functions a dynamic part of algebra.