A quadratic function is typically expressed in the form of:
f(x) = a(x – r1)(x – r2)
where r1 and r2 are the roots (or zeros) of the function, and a is a non-zero constant that affects the vertical stretch or compression of the graph.
In your case, the real zeros are given as x = 4 and x = 1. We can substitute these values into the formula:
r1 = 4 and r2 = 1,
The quadratic function will then be:
f(x) = a(x – 4)(x – 1)
Expanding this expression, we get:
f(x) = a[(x – 4)(x – 1)]
f(x) = a[x^2 – 5x + 4]
The most basic version of this function, where a = 1, is:
f(x) = x^2 – 5x + 4
This function will have its parabola opening upwards with the vertex located between the zeros at x = 4 and x = 1. The exact shape of the parabola can vary depending on the value of a. If a is positive, the parabola opens upwards, while if a is negative, it opens downwards.
In summary, a quadratic function with real zeros at x = 4 and x = 1 can be expressed in various forms depending on the value of a, but one instance is:
f(x) = x^2 – 5x + 4
Feel free to adjust the constant a to fit your specific needs!