Writing a Quadratic Function, Given its Zeros
Write a quadratic function given its zeros using the factored form f(x)=a(x-r1)(x-r2): expand to standard form and determine the leading coefficient from additional given information.
Key Concepts
To write a quadratic function from its zeros, reverse the solving process. If the zeros are $r 1$ and $r 2$, start with the factored equation $(x r 1)(x r 2) = 0$ and expand it into the standard form $ax^2 + bx + c = 0$.
Write a function with zeros $4$ and $ 6$. Start with the factors: $(x 4)(x ( 6))=0$, which is $(x 4)(x+6)=0$. Expand it: $x^2 + 6x 4x 24 = 0$. The function is $f(x) = x^2 + 2x 24$. Write a function with zeros $ \frac{3}{2}$ and $1$. Start with $(x+\frac{3}{2})(x 1)=0$. Multiply by 2 to clear the fraction: $(2x+3)(x 1)=0$. Expand: $2x^2 2x + 3x 3 = 0$. The function is $f(x)=2x^2+x 3$.
This is like being a math magician who builds a quadratic from scratch! If you know the 'ending'—the zeros where the graph hits the x axis—you can work backward. Just turn those zeros back into factors, multiply them together, and poof! You have perfectly reconstructed the original quadratic function. It’s a reverse puzzle that is super satisfying to solve.
Common Questions
How do you write a quadratic function given its zeros?
If the zeros are r1 and r2, write the factored form f(x)=a(x-r1)(x-r2) where a is any nonzero real number. To determine a, substitute any additional known point such as the y-intercept or vertex into the equation and solve for a. Expand to get standard form if needed.
How do you write a quadratic with zeros at x=3 and x=-5 that passes through (0,30)?
Start with f(x)=a(x-3)(x+5). Substitute (0,30): 30=a(0-3)(0+5)=a(-3)(5)=-15a. Solve: a=-2. The function is f(x)=-2(x-3)(x+5). Expand: f(x)=-2(x^2+2x-15)=-2x^2-4x+30.
Can a quadratic have only one zero?
Yes. When the two zeros are equal (r1=r2=r), the quadratic has a double zero and factors as f(x)=a(x-r)^2. The graph touches the x-axis at x=r but does not cross it. This corresponds to a zero of multiplicity 2.