Define the quadratic function ƒ having x-intercepts (2, 0) and (5, 0) and y-intercept (0, 5).

Verified step by step guidance

1. To define a quadratic function with given x-intercepts, use the factored form: f(x) = a(x - x_1)(x - x_2), where x_1 and x_2 are the x-intercepts.
2. Substitute the given x-intercepts (2, 0) and (5, 0) into the factored form: f(x) = a(x - 2)(x - 5).
3. Use the y-intercept (0, 5) to find the value of 'a'. Substitute x = 0 and f(x) = 5 into the equation: 5 = a(0 - 2)(0 - 5).
4. Simplify the equation to solve for 'a': 5 = a(-2)(-5).
5. Once 'a' is found, substitute it back into the equation f(x) = a(x - 2)(x - 5) to define the quadratic function.

Key Concepts

Quadratic Function

A quadratic function is a polynomial function of degree two, typically expressed in the form ƒ(x) = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of 'a'. Understanding the general form is essential for defining specific quadratic functions based on given intercepts.

X-Intercepts

X-intercepts are the points where a function crosses the x-axis, meaning the output (y-value) is zero. For a quadratic function, if the x-intercepts are known, they can be used to express the function in factored form as ƒ(x) = a(x - r1)(x - r2), where r1 and r2 are the x-intercepts. In this case, the x-intercepts (2, 0) and (5, 0) will help in constructing the quadratic function.

Y-Intercept

The y-intercept of a function is the point where the graph intersects the y-axis, which occurs when x = 0. For a quadratic function, the y-intercept can be found by evaluating the function at x = 0, resulting in the constant term 'c' in the standard form. In this problem, the y-intercept (0, 5) provides a specific value that helps determine the coefficient 'a' when combined with the x-intercepts.