When you look at quadratic graphs, like the familiar equation (y = ax^2 + bx + c), it's really interesting to see how stretches (or squeezes) can change how the graph looks. Understanding this can help you in your maths exams and also help you see the beauty in these changes.
In math, a stretch means making the graph bigger or smaller, either away from or towards the x-axis (side to side) or y-axis (up and down). For quadratic graphs, we mainly talk about two types of stretches:
Let’s begin with vertical stretches. If you have a quadratic function like (y = ax^2), changing the number (a) can make the graph steeper or flatter:
If (|a| > 1): The graph becomes narrower (it stretches up). For example, if you change (y = x^2) to (y = 2x^2), the graph looks steeper and more “compressed” compared to the regular shape.
If (0 < |a| < 1): The graph widens (it squeezes down). So, if you change (y = x^2) to (y = \frac{1}{2}x^2), it looks wider; it appears more spread out.
Horizontal stretches can be a bit harder to picture! They happen when we change (x) in the function. For example:
If you have (y = (x-1)^2) and change (x) to (kx), making it (y = (kx-1)^2), it changes how wide the graph looks depending on (k):
If (|k| > 1): The graph gets narrower (it compresses side to side). For instance, (y = (2x-1)^2) looks thinner than (y = (x-1)^2).
If (0 < |k| < 1): The graph gets wider (it stretches side to side). So, for (y = (\frac{1}{2}x-1)^2), the parabola spreads out more.
Here’s a quick recap of how stretches change quadratic graphs:
It really helps to draw these graphs or use graphing programs to see the differences. You can easily see how changing (a) and (k) can make a regular parabola look quite different.
Overall, understanding these stretches not only helps you with homework and tests but also gives you a better feel for how quadratic functions behave. You might start to enjoy “playing” with these graphs, almost like being an artist using math to change and redesign how graphs look. Keep trying different numbers, and you’ll see how fun and interesting math can be!
When you look at quadratic graphs, like the familiar equation (y = ax^2 + bx + c), it's really interesting to see how stretches (or squeezes) can change how the graph looks. Understanding this can help you in your maths exams and also help you see the beauty in these changes.
In math, a stretch means making the graph bigger or smaller, either away from or towards the x-axis (side to side) or y-axis (up and down). For quadratic graphs, we mainly talk about two types of stretches:
Let’s begin with vertical stretches. If you have a quadratic function like (y = ax^2), changing the number (a) can make the graph steeper or flatter:
If (|a| > 1): The graph becomes narrower (it stretches up). For example, if you change (y = x^2) to (y = 2x^2), the graph looks steeper and more “compressed” compared to the regular shape.
If (0 < |a| < 1): The graph widens (it squeezes down). So, if you change (y = x^2) to (y = \frac{1}{2}x^2), it looks wider; it appears more spread out.
Horizontal stretches can be a bit harder to picture! They happen when we change (x) in the function. For example:
If you have (y = (x-1)^2) and change (x) to (kx), making it (y = (kx-1)^2), it changes how wide the graph looks depending on (k):
If (|k| > 1): The graph gets narrower (it compresses side to side). For instance, (y = (2x-1)^2) looks thinner than (y = (x-1)^2).
If (0 < |k| < 1): The graph gets wider (it stretches side to side). So, for (y = (\frac{1}{2}x-1)^2), the parabola spreads out more.
Here’s a quick recap of how stretches change quadratic graphs:
It really helps to draw these graphs or use graphing programs to see the differences. You can easily see how changing (a) and (k) can make a regular parabola look quite different.
Overall, understanding these stretches not only helps you with homework and tests but also gives you a better feel for how quadratic functions behave. You might start to enjoy “playing” with these graphs, almost like being an artist using math to change and redesign how graphs look. Keep trying different numbers, and you’ll see how fun and interesting math can be!