Writing a Quadratic Function from Its Zeros
Writing a quadratic function from its zeros in Algebra 1 (California Reveal Math, Grade 9) uses the factored form: if the zeros are r₁ and r₂, then f(x) = a(x - r₁)(x - r₂), where a is the leading coefficient. To find a, substitute a known point on the parabola and solve. For example, zeros at x = 2 and x = -3 give f(x) = a(x - 2)(x + 3). If a = 1 and the parabola passes through (0, -6), then a = 1 and f(x) = x² + x - 6. This skill connects roots, factors, and function rules in Algebra 1 quadratic functions.
Key Concepts
If a quadratic function has zeros (roots) $r 1$ and $r 2$, it can be written in factored form as:.
$$f(x) = a(x r 1)(x r 2)$$.
Common Questions
How do you write a quadratic function from its zeros?
Use factored form: f(x) = a(x - r₁)(x - r₂), where r₁ and r₂ are the zeros. If another point is known, substitute it to find a.
What are the zeros of a quadratic function?
The zeros (also called roots or x-intercepts) are the x-values where f(x) = 0. They are where the parabola crosses or touches the x-axis.
What is the leading coefficient a in the factored form?
The leading coefficient a determines the vertical stretch/compression and whether the parabola opens up (a > 0) or down (a < 0). Without additional information, a = 1 is assumed.
How do zeros relate to factors of a quadratic?
If r is a zero, then (x - r) is a factor. This follows from the Factor Theorem: (x - r) divides the polynomial evenly if and only if r is a root.
Where is writing a quadratic from its zeros covered in California Reveal Math Algebra 1?
This skill is taught in California Reveal Math, Algebra 1, as part of Grade 9 quadratic functions and their representations.
Can a quadratic have only one zero?
Yes, if the parabola touches but does not cross the x-axis (a repeated root). For example, f(x) = (x - 3)² = 0 has only x = 3 as a zero.
How is this related to expanding vs. factoring quadratics?
Writing from zeros uses factored form. Multiplying out the factors gives standard form. These are inverse operations and both are important skills in Algebra 1.