Question

Write a quadratic function in the form \(f(x)=x^2+b x+c\) that has zeros 8 and 11.

Short answer

The quadratic function in the form \(f(x) = x^2 + bx + c\) having zeros at 8 and 11 is \(f(x) = x^2 - 19x + 88 \).

Step-by-step solution

Calculate sums and products

Calculate the sum and the product of the roots. The sum of roots equals 8+11=19 and the product of roots equals 8*11=88.

Write the function

Now, you can use these results to identify the coefficients b and c in the quadratic function since b should be the opposite of the sum of roots and c equals the product of roots. The function \(f(x) = x^2 - b x + c\) can be written as \(f(x) = x^2 - 19x + 88 \).

Key concepts

Sum and Product of Roots

Understanding the relationship between the roots of a quadratic function and its coefficients is crucial when dealing with algebraic functions like the one in our exercise. The roots, commonly referred to as zeros, are the values of 'x' for which the quadratic function equals zero. The 'sum and product of roots' is a fundamental concept which states that for a quadratic equation of the form \(ax^2+bx+c=0\), the sum of the roots \(\alpha\) and \(\beta\) is equal to \(-b/a\) and their product is \(c/a\).

In the given exercise, we used these relationships to find the coefficients of the quadratic function with zeros 8 and 11. By calculating the sum, \(8+11=19\), and product, \(8 \times 11=88\), we were able to determine that in the function \(f(x)=x^2+bx+c\), the value of 'b' must be the negative of the sum of roots, and 'c' must be the product of the roots. Therefore, 'b' is \(-19\) and 'c' is \(88\), leading to the function \(f(x)=x^2-19x+88\). This insight is immensely helpful for quickly constructing quadratic equations from given roots.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x with a non-zero coefficient of \(x^2\). It has the standard form \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' represent constants with 'a' ≠ 0. The solutions to a quadratic equation are called its 'roots' and can be real or complex numbers.

To solve a quadratic equation, one can use different methods such as factoring, completing the square, or applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). The discriminant \(\Delta = b^2 - 4ac\) in the quadratic formula indicates the nature of the roots. If \(\Delta > 0\), the roots are distinct and real; if \(\Delta = 0\), there is one real root (a repeated root); if \(\Delta < 0\), the roots are complex and occur in conjugate pairs. In our exercise improvement advice, it's essential to ensure students understand how these concepts apply when they reverse the process to write the function starting from the roots.

Algebraic Functions

An algebraic function is a type of function that can be represented by a polynomial equation of finite order. Each term in an algebraic function is constituted of a constant or a product of constants and variables raised to a whole number exponent. The polynomial functions, particularly quadratic functions, are some of the simplest examples of algebraic functions.

In our example, the quadratic function \(f(x) = x^2 - 19x + 88\) is an algebraic function with real coefficients. When graphed, it will typically show a parabolic shape, opening upwards since the coefficient of \(x^2\) is positive. The location of its roots on the x-axis is crucial, as they represent the x-values where the function crosses the x-axis, corresponding to the zeros 8 and 11 in our exercise. Helping students visualize this can be instrumental in their understanding of how algebraic functions behave and how the coefficients affect the shape and position of the graph.