Which Function Has Zeros At X 10 And X 2

Which Function Has Zeros at (x = 10) and (x = 2)?

When you hear that a function “has zeros at (x = 10) and (x = 2),” you’re being told exactly where the graph of that function crosses the horizontal axis. In algebraic terms, a zero (or root) of a function (f(x)) is a value of (x) for which (f(x) = 0). So, if a function has zeros at (x = 10) and (x = 2), it satisfies the two equations

Real talk — this step gets skipped all the time Turns out it matters..

[ f(10) = 0 \quad\text{and}\quad f(2) = 0 . ]

The simplest class of functions that can meet these conditions is the set of polynomials, because any polynomial that factors into linear terms will have a zero at each factor’s root. Let’s explore how to build such a polynomial, why the quadratic form is the minimal choice, and how the shape of the graph is influenced by the choice of the leading coefficient.

1. Building the Polynomial from Its Zeros

1.1. The Factor Theorem

The Factor Theorem states that if (r) is a zero of a polynomial (P(x)), then ((x - r)) is a factor of (P(x)). Applying this to our two zeros:

• Since (x = 10) is a zero, ((x - 10)) is a factor.
• Since (x = 2) is a zero, ((x - 2)) is a factor.

Multiplying these factors gives the minimal polynomial that has the required zeros:

[ P(x) = a,(x - 10)(x - 2), ]

where (a) is a non‑zero constant that scales the graph vertically but does not change the zeros Simple, but easy to overlook..

1.2. Expanding the Expression

Expanding the product yields a more familiar quadratic form:

[ \begin{aligned} P(x) &= a,(x - 10)(x - 2) \ &= a,(x^2 - 12x + 20) \ &= a,x^2 - 12a,x + 20a . \end{aligned} ]

Thus, every quadratic function whose zeros are (10) and (2) can be written as

[ f(x) = a,x^2 - 12a,x + 20a . ]

The parameter (a) determines whether the parabola opens upward ((a > 0)) or downward ((a < 0)), and how steeply it rises or falls.

2. Choosing the Leading Coefficient (a)

2.1. The Standard Form

The most common choice is (a = 1), giving the standard quadratic:

[ f(x) = (x - 10)(x - 2) = x^2 - 12x + 20 . ]

This function is often used as a textbook example because it is the simplest polynomial that satisfies the zero conditions And that's really what it comes down to. Nothing fancy..

2.2. Scaling the Graph

If you multiply the entire function by a constant (k) (i.e., set (a = k)), the graph stretches or compresses vertically:

• (k > 1): The parabola becomes steeper; values far from the zeros grow faster.
• (0 < k < 1): The parabola flattens; the graph is less steep.
• (k < 0): The parabola flips upside down; the zeros remain the same but the opening direction reverses.

For example:

• (f(x) = 2x^2 - 24x + 40) (here (a = 2)) opens upward and is twice as steep. Which means - (f(x) = -0. 5x^2 + 6x - 10) (here (a = -0.5)) opens downward and is half as steep.

2.3. Intercept and Vertex

The y‑intercept is found by evaluating (f(0)):

[ f(0) = a(0 - 10)(0 - 2) = a \times 20 = 20a . ]

Thus, the y‑intercept is directly proportional to (a). The vertex of the parabola lies midway between the zeros, at

[ x_{\text{vertex}} = \frac{10 + 2}{2} = 6. ]

Plugging (x = 6) into the function gives the y‑coordinate of the vertex:

[ f(6) = a(6 - 10)(6 - 2) = a(-4)(4) = -16a . ]

So the vertex is at ((6, -16a)). If (a > 0), the vertex is a minimum point; if (a < 0), it is a maximum point.

3. Graphical Interpretation

A quick sketch for (a = 1) shows a parabola opening upward, crossing the x‑axis at 2 and 10, dipping to a minimum at ((6, -16)), and intersecting the y‑axis at ((0, 20)).

4. Extending Beyond Quadratics

While a quadratic is the minimal polynomial that satisfies two distinct zeros, you can construct higher‑degree polynomials that also have the same zeros by multiplying the quadratic factor by additional terms that do not introduce new zeros (or introduce new zeros elsewhere). For instance:

[ f(x) = (x - 10)(x - 2)(x^2 + 1) . ]

Here, (x^2 + 1) never equals zero for real (x), so the only real zeros remain at 10 and 2. The graph will have the same x‑intercepts but will exhibit additional curvature due to the quartic term Turns out it matters..

5. Practical Applications

5.1. Engineering

In control systems, the zeros of a transfer function determine the system’s frequency response. Knowing that a system has zeros at (x = 10) and (x = 2) helps engineers predict attenuation at those frequencies.

5.2. Physics

In projectile motion, the quadratic equation (y = ax^2 + bx + c) often represents the trajectory. Setting the zeros to specific values can model a projectile that lands at two distinct points along the x‑axis.

5.3. Finance

Quadratic cost functions with zeros at particular production levels can represent economies of scale or break‑even points in cost‑benefit analyses.

6. Frequently Asked Questions

Real talk — this step gets skipped all the time.

7. Conclusion

A function that has zeros at (x = 10) and (x = 2) is most naturally described by a quadratic polynomial of the form

[ f(x) = a,(x - 10)(x - 2) = a,x^2 - 12a,x + 20a , ]

where (a) is any non‑zero constant. Which means the parameter (a) controls the vertical stretch and the direction in which the parabola opens, while the zeros themselves are fixed by the linear factors. This concise form not only satisfies the zero conditions but also provides a clear geometric picture: the graph crosses the x‑axis at 2 and 10, reaches a vertex at ((6, -16a)), and intersects the y‑axis at ((0, 20a)). Whether used in algebra problems, engineering models, or real‑world applications, this function exemplifies how the roots of a polynomial dictate its fundamental behavior.