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

Verified step by step guidance

1. Recall that a quadratic function with x-intercepts at \( x = a \) and \( x = b \) can be written in factored form as \( f(x) = k(x - a)(x - b) \), where \( k \) is a constant.
2. Substitute the given x-intercepts into the factored form: \( f(x) = k(x - 2)(x - 5) \).
3. Use the y-intercept \( (0, 5) \) to find the value of \( k \). Substitute \( x = 0 \) and \( f(0) = 5 \) into the equation: \( 5 = k(0 - 2)(0 - 5) \).
4. Simplify the right side: \( 5 = k(-2)(-5) = 10k \).
5. Solve for \( k \) by dividing both sides by 10: \( k = \frac{5}{10} \). Then write the final quadratic function as \( f(x) = \frac{5}{10}(x - 2)(x - 5) \).

Key Concepts

Quadratic Function and Standard Form

A quadratic function is a polynomial of degree two, typically written as f(x) = ax² + bx + c. Understanding this form helps in identifying the coefficients and how they affect the graph's shape and position.

X-Intercepts and Factored Form

X-intercepts are points where the graph crosses the x-axis, meaning f(x) = 0. Knowing the x-intercepts allows you to write the quadratic in factored form as f(x) = a(x - r1)(x - r2), where r1 and r2 are the roots.

Y-Intercept and Finding the Leading Coefficient

The y-intercept is the point where the graph crosses the y-axis (x=0). Using the y-intercept value helps determine the leading coefficient 'a' in the factored form by substituting x=0 and solving for 'a'.