Understanding the Function with Given X-Intercepts
To find a function that has x-intercepts at the points (0, 0) and (4, 0), we need to consider how x-intercepts are defined. X-intercepts occur where the function crosses the x-axis, which means at these points, the function’s value is equal to zero.
Constructing the Function
Since the function must equal zero at both (0, 0) and (4, 0), we can represent it in factored form:
f(x) = a * x * (x – 4)
Here, a is a constant that can be any non-zero value which will stretch or compress the graph vertically, but does not affect the x-intercepts.
Example of the Function
Setting a to 1 for simplicity, we have:
f(x) = x * (x – 4)
This expands to:
f(x) = x^2 – 4x
Now, solving for the x-intercepts:
1. Set f(x) = 0:
x^2 – 4x = 0
2. Factor out an x:
x(x – 4) = 0
3. This gives us two solutions:
x = 0 and x = 4
Conclusion
The quadratic function f(x) = x^2 – 4x has the desired x-intercepts at (0, 0) and (4, 0). If you adjust the value of a (for example, setting a = 2), you’d still have the same x-intercepts, but the shape of the graph may change. This gives you a versatile way to create functions that meet your criteria while also offering a variety of behaviors. Enjoy experimenting with different values for a!