Defining Zeros of a Function
The zeros of a function are the x-values where f(x) = 0 — also called roots or solutions — a critical concept for quadratic equations in enVision Algebra 1 Chapter 9 for Grade 11. For f(x) = x² - 9, x = 3 is a zero because f(3) = 9 - 9 = 0. For g(x) = x² - 5x + 6, both x = 2 and x = 3 are zeros because g(2) = 0 and g(3) = 0. On a graph, zeros are the x-intercepts — the points where the parabola crosses the horizontal axis. Finding zeros is equivalent to solving the quadratic equation ax² + bx + c = 0.
Key Concepts
Property The zeros of a function $f(x)$ are the values of $x$ for which $f(x) = 0$. These values are also the solutions, or roots, of the equation $f(x) = 0$. For a quadratic function $f(x) = ax^2 + bx + c$, the zeros are the solutions to the quadratic equation $ax^2 + bx + c = 0$.
Examples If $x = 3$ is a zero of the function $f(x) = x^2 9$, then $f(3) = 3^2 9 = 9 9 = 0$. The zeros of the function $g(x) = x^2 5x + 6$ are $x = 2$ and $x = 3$, because $g(2)=0$ and $g(3)=0$. These are the solutions to the equation $x^2 5x + 6 = 0$. On a graph, the zeros of a function are the x coordinates of the points where the graph intersects the x axis, also known as the x intercepts.
Explanation A "zero" of a function is an input value that results in an output of zero. Finding the zeros of a quadratic function is the same as solving the related quadratic equation. Graphically, the real zeros of a function correspond to the x intercepts of its graph. This is because any point on the x axis has a y coordinate of zero.
Common Questions
What are the zeros of a function?
The zeros are the x-values that make f(x) = 0. They are also called roots or solutions and correspond to x-intercepts on the graph.
How do you verify that x = 3 is a zero of f(x) = x² - 9?
Substitute x = 3: f(3) = 3² - 9 = 9 - 9 = 0. Since the output is 0, x = 3 is confirmed as a zero.
What are the zeros of g(x) = x² - 5x + 6?
x = 2 and x = 3. Check: g(2) = 4 - 10 + 6 = 0 and g(3) = 9 - 15 + 6 = 0. These are the x-intercepts of the parabola.
How are zeros of a function related to x-intercepts on a graph?
They are the same thing. Zeros are the x-values where the function output is 0, which geometrically means the graph crosses or touches the x-axis.
How many zeros can a quadratic function have?
A quadratic can have 0, 1, or 2 real zeros. This depends on the discriminant: two distinct zeros if b²-4ac > 0, one repeated zero if b²-4ac = 0, and no real zeros if b²-4ac < 0.