# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/quadratic-function/quadratic-function-construction
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/quadratic-function/quadratic-function-construction/en.mdx

Output docs content for large language models.

---

import { LineEquation } from "@repo/design-system/components/contents/line-equation";
import { getColor } from "@repo/design-system/lib/color";

export const metadata = {
  title: "Constructing Quadratic Functions",
  description: "Learn to construct quadratic functions using three points, vertex form, roots, and symmetry. Master different methods with step-by-step examples and solutions.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/19/2025",
  subject: "Quadratic Functions",
};

## How to Determine Quadratic Function Equations

Constructing a quadratic function means determining the form of the equation <InlineMath math="f(x) = ax^2 + bx + c" /> based on given information. There are several different ways to construct a quadratic function, depending on the available information.

## Forms of Quadratic Functions

Before we start constructing quadratic functions, let's understand the three forms of quadratic functions:

1. **Standard Form**: <InlineMath math="f(x) = ax^2 + bx + c" />
2. **Factored Form**: <InlineMath math="f(x) = a(x - p)(x - q)" /> where p and q are the roots of the equation
3. **Vertex Form**: <InlineMath math="f(x) = a(x - h)^2 + k" /> where (h, k) is the vertex point

These three forms are interrelated and can be converted from one form to another.

## Methods of Constructing Quadratic Functions

### Three Points

The most common way to construct a quadratic function is by using three known points on the curve.

If we have three points <InlineMath math="(x_1, y_1)" />, <InlineMath math="(x_2, y_2)" />, and <InlineMath math="(x_3, y_3)" />, we can substitute these values into the standard equation to get three linear equations with three variables a, b, and c.

<div className="flex flex-col gap-4">
  <BlockMath math="y_1 = ax_1^2 + bx_1 + c" />
  <BlockMath math="y_2 = ax_2^2 + bx_2 + c" />
  <BlockMath math="y_3 = ax_3^2 + bx_3 + c" />
</div>

**Example**:

Find the quadratic function that passes through points K(-1, 0), L(0, -3), and M(1, -4).

Substitute the coordinate values into the standard equation.

<div className="flex flex-col gap-4">
  <BlockMath math="K(-1, 0): 0 = a(-1)^2 + b(-1) + c" />
  <BlockMath math="L(0, -3): -3 = a(0)^2 + b(0) + c" />
  <BlockMath math="M(1, -4): -4 = a(1)^2 + b(1) + c" />
</div>

Simplify these equations.

<div className="flex flex-col gap-4">
  <BlockMath math="0 = a - b + c" />
  <BlockMath math="-3 = c" />
  <BlockMath math="-4 = a + b + c" />
</div>

Substitute <InlineMath math="c = -3" /> into the first and third equations.

<div className="flex flex-col gap-4">
  <BlockMath math="0 = a - b - 3" />
  <BlockMath math="-4 = a + b - 3" />
</div>

Simplify and add the two equations to find the value of a.

<div className="flex flex-col gap-4">
  <BlockMath math="a - b = 3" />
  <BlockMath math="a + b = -1" />
  <BlockMath math="2a = 2" />
  <BlockMath math="a = 1" />
</div>

Substitute the value of a into the equation <InlineMath math="a - b = 3" /> to find the value of b.

<div className="flex flex-col gap-4">
  <BlockMath math="1 - b = 3" />
  <BlockMath math="b = -2" />
</div>

The resulting quadratic function is:

<BlockMath math="f(x) = x^2 - 2x - 3" />

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = x^2 - 2x - 3" />
      </>
    }
    description={
      <>
        Parabola passing through points <InlineMath math="K(-1, 0)" />,{" "}
        <InlineMath math="L(0, -3)" />, and <InlineMath math="M(1, -4)" />.
      </>
    }
    cameraPosition={[2, 3, 8]}
    data={[
      {
        points: Array.from({ length: 7 }, (_, i) => {
          const x = i - 3; // x values from -3 to 3
          return { x, y: x * x - 2 * x - 3, z: 0 };
        }),
        color: getColor("TEAL"),
        labels: [
          {
            text: "f(x) = x² - 2x - 3",
            at: 5,
            offset: [2, 0.3, 0],
          },
          {
            text: "L(0, -3)",
            at: 3,
            offset: [-1, 0, 0],
          },
          {
            text: "M(1, -4)",
            at: 4,
            offset: [0, -0.5, 0],
          },
          {
            text: "K(-1, 0)",
            at: 2,
            offset: [-1, 0.3, 0],
          },
        ],
      },
    ]}
  />
</div>

### Vertex Point

If we know the coordinates of the vertex (h, k) and another point (p, q) on the graph, we can use the vertex form <InlineMath math="f(x) = a(x - h)^2 + k" /> to find the value of a.

**Example**:

Find the quadratic function that has a vertex at (2, 0) and passes through the point (4, 4).

Use the vertex form <InlineMath math="f(x) = a(x - h)^2 + k" /> with h = 2 and k = 0.

<BlockMath math="f(x) = a(x - 2)^2 + 0 = a(x - 2)^2" />

Substitute the point (4, 4) to find the value of a.

<div className="flex flex-col gap-4">
  <BlockMath math="4 = a(4 - 2)^2" />
  <BlockMath math="4 = 4a" />
  <BlockMath math="a = 1" />
</div>

The resulting quadratic function is:

<BlockMath math="f(x) = (x - 2)^2 = x^2 - 4x + 4" />

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = (x - 2)^2" />
      </>
    }
    description={
      <>
        Parabola with vertex at <InlineMath math="(2, 0)" /> and passing through
        point <InlineMath math="(4, 4)" />.
      </>
    }
    cameraPosition={[6, 6, 8]}
    data={[
      {
        points: Array.from({ length: 9 }, (_, i) => {
          const x = i - 1; // x values from -1 to 7
          return { x, y: Math.pow(x - 2, 2), z: 0 };
        }),
        color: getColor("INDIGO"),
        labels: [
          {
            text: "f(x) = (x - 2)²",
            at: 7,
            offset: [0.3, 0.3, 0],
          },
          {
            text: "(2, 0)",
            at: 3,
            offset: [0, 0.3, 0],
          },
          {
            text: "(4, 4)",
            at: 5,
            offset: [0.3, 0.3, 0],
          },
        ],
      },
    ]}
  />
</div>

### Equation Roots

If we know the roots (x-intercepts) p and q of the quadratic function, and an additional point (r, s) on the curve, we can use the factored form <InlineMath math="f(x) = a(x - p)(x - q)" />.

**Example**:

Find the quadratic function that has roots at x = -2 and x = 3, and passes through the point (1, -6).

Use the factored form <InlineMath math="f(x) = a(x - p)(x - q)" /> with p = -2 and q = 3.

<BlockMath math="f(x) = a(x - (-2))(x - 3) = a(x + 2)(x - 3)" />

Substitute the point (1, -6) to find the value of a.

<div className="flex flex-col gap-4">
  <BlockMath math="-6 = a(1 + 2)(1 - 3)" />
  <BlockMath math="-6 = a(3)(-2)" />
  <BlockMath math="-6 = -6a" />
  <BlockMath math="a = 1" />
</div>

The resulting quadratic function is:

<BlockMath math="f(x) = (x + 2)(x - 3) = x^2 - x - 6" />

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = (x + 2)(x - 3)" />
      </>
    }
    description={
      <>
        Parabola with roots at <InlineMath math="x = -2" /> and{" "}
        <InlineMath math="x = 3" />, passing through point{" "}
        <InlineMath math="(1, -6)" />.
      </>
    }
    cameraPosition={[4, 5, 10]}
    data={[
      {
        points: Array.from({ length: 11 }, (_, i) => {
          const x = i - 4; // x values from -4 to 6
          return { x, y: (x + 2) * (x - 3), z: 0 };
        }),
        color: getColor("ORANGE"),
        labels: [
          {
            text: "f(x) = (x + 2)(x - 3)",
            at: 6,
            offset: [3, 0.3, 0],
          },
          {
            text: "(-2, 0)",
            at: 2,
            offset: [0.3, 0.3, 0],
          },
          {
            text: "(3, 0)",
            at: 7,
            offset: [0.3, 0.3, 0],
          },
          {
            text: "(1, -6)",
            at: 5,
            offset: [0.3, -0.3, 0],
          },
        ],
      },
    ]}
  />
</div>

### Axis of Symmetry and Discriminant

We can also construct a quadratic function by knowing the axis of symmetry (or the x-coordinate of the vertex) and the value of the discriminant.

**Example**:

Find a quadratic function with axis of symmetry x = 1 and discriminant D = 16.

From the axis of symmetry x = 1, we know that <InlineMath math="-\frac{b}{2a} = 1" />, so <InlineMath math="b = -2a" />.

From the discriminant D = 16, we know that <InlineMath math="b^2 - 4ac = 16" />.

Substitute <InlineMath math="b = -2a" /> into the discriminant equation.

<div className="flex flex-col gap-4">
  <BlockMath math="(-2a)^2 - 4ac = 16" />
  <BlockMath math="4a^2 - 4ac = 16" />
  <BlockMath math="4a(a - c) = 16" />
  <BlockMath math="a(a - c) = 4" />
</div>

There are many values of a and c that satisfy this equation. Let's take the simple case with a = 1.

<div className="flex flex-col gap-4">
  <BlockMath math="1(1 - c) = 4" />
  <BlockMath math="1 - c = 4" />
  <BlockMath math="c = -3" />
</div>

With a = 1, b = -2, and c = -3, the resulting quadratic function is:

<BlockMath math="f(x) = x^2 - 2x - 3" />

### Symmetric Coordinates

We can construct a quadratic function by utilizing the symmetry property of parabolas.

**Example**:

Determine the quadratic function that passes through the points (0, 0), (4, 1), and (-4, 1).

Since the points (4, 1) and (-4, 1) have the same y-value and are at the same distance from the y-axis, the curve is symmetric about the y-axis. This means the axis of symmetry is x = 0, and the vertex is at (0, 0).

Use the vertex form <InlineMath math="f(x) = a(x - h)^2 + k" /> with h = 0 and k = 0.

<BlockMath math="f(x) = ax^2" />

Substitute the point (4, 1) to find the value of a.

<div className="flex flex-col gap-4">
  <BlockMath math="1 = a(4)^2" />
  <BlockMath math="1 = 16a" />
  <BlockMath math="a = \frac{1}{16}" />
</div>

The resulting quadratic function is:

<BlockMath math="f(x) = \frac{1}{16}x^2" />

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = \frac{1}{16}x^2" />
      </>
    }
    description={
      <>
        Parabola passing through points <InlineMath math="(0, 0)" />,{" "}
        <InlineMath math="(4, 1)" />, and <InlineMath math="(-4, 1)" />.
      </>
    }
    data={[
      {
        points: Array.from({ length: 13 }, (_, i) => {
          const x = i - 6; // x values from -6 to 6
          return { x, y: (1 / 16) * x * x, z: 0 };
        }),
        color: getColor("TEAL"),
        labels: [
          {
            text: "f(x) = (1/16)x²",
            at: 11,
            offset: [-2, 0.3, 0],
          },
          {
            text: "(0, 0)",
            at: 6,
            offset: [0.3, 0.3, 0],
          },
          {
            text: "(4, 1)",
            at: 10,
            offset: [0.3, 0.3, 0],
          },
          {
            text: "(-4, 1)",
            at: 2,
            offset: [0.3, 0.3, 0],
          },
        ],
      },
    ]}
  />
</div>

## Transformations Between Quadratic Function Forms

### Standard Form to Vertex Form

To convert <InlineMath math="f(x) = ax^2 + bx + c" /> to vertex form <InlineMath math="f(x) = a(x - h)^2 + k" />:

1. Determine the x-coordinate of the vertex: <InlineMath math="h = -\frac{b}{2a}" />
2. Calculate the function value at the vertex: <InlineMath math="k = f(h) = f(-\frac{b}{2a})" />
3. Or use the formula: <InlineMath math="k = c - \frac{b^2}{4a}" />

### Standard Form to Factored Form

To convert <InlineMath math="f(x) = ax^2 + bx + c" /> to factored form <InlineMath math="f(x) = a(x - p)(x - q)" />:

1. Determine the roots of the equation <InlineMath math="ax^2 + bx + c = 0" /> using the formula:

   <BlockMath math="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}" />

2. If the roots are p and q, then:

   <BlockMath math="f(x) = a(x - p)(x - q)" />

### Factored Form to Standard Form

To convert <InlineMath math="f(x) = a(x - p)(x - q)" /> to standard form <InlineMath math="f(x) = ax^2 + bx + c" />:

<div className="flex flex-col gap-4">
  <BlockMath math="f(x) = a(x - p)(x - q)" />
  <BlockMath math="f(x) = a(x^2 - qx - px + pq)" />
  <BlockMath math="f(x) = a(x^2 - (p+q)x + pq)" />
  <BlockMath math="f(x) = ax^2 - a(p+q)x + apq" />
</div>

By comparing with the standard form, we get:

- <InlineMath math="b = -a(p+q)" />
- <InlineMath math="c = apq" />

## Exercises and Solutions

### Exercise 1

Find the quadratic function that passes through the points (-2, 4), (1, -5), and (3, 7).

**Answer**:

Substitute the points into the standard equation:

<div className="flex flex-col gap-4">
  <BlockMath math="4 = a(-2)^2 + b(-2) + c = 4a - 2b + c" />
  <BlockMath math="-5 = a(1)^2 + b(1) + c = a + b + c" />
  <BlockMath math="7 = a(3)^2 + b(3) + c = 9a + 3b + c" />
</div>

From the second equation, <InlineMath math="c = -5 - a - b" />.
Substitute into the first equation, then solve:

<div className="flex flex-col gap-4">
  <BlockMath math="4 = 4a - 2b + (-5 - a - b)" />
  <BlockMath math="9 = 3a - 3b" />
  <BlockMath math="3 = a - b" />
  <BlockMath math="a = 3 + b" />
</div>

Substitute into the third equation:

<div className="flex flex-col gap-4">
  <BlockMath math="7 = 9(3 + b) + 3b + (-5 - (3 + b) - b)" />
  <BlockMath math="7 = 19 + 10b" />
  <BlockMath math="b = -\frac{12}{10} = -\frac{6}{5}" />
</div>

Then:

<div className="flex flex-col gap-4">
  <BlockMath math="a = 3 + (-\frac{6}{5}) = \frac{9}{5}" />
  <BlockMath math="c = -5 - \frac{9}{5} - (-\frac{6}{5}) = -\frac{28}{5}" />
</div>

Therefore, the quadratic function is:

<BlockMath math="f(x) = \frac{9}{5}x^2 - \frac{6}{5}x - \frac{28}{5}" />

### Exercise 2

Determine the quadratic function that has a vertex at (-1, 4) and passes through the point (2, -5).

**Answer**:

Use the vertex form with h = -1 and k = 4:

<BlockMath math="f(x) = a(x - (-1))^2 + 4 = a(x + 1)^2 + 4" />

Substitute the point (2, -5):

<div className="flex flex-col gap-4">
  <BlockMath math="-5 = a(2 + 1)^2 + 4" />
  <BlockMath math="-5 = 9a + 4" />
  <BlockMath math="-9 = 9a" />
  <BlockMath math="a = -1" />
</div>

Therefore, the quadratic function is:

<BlockMath math="f(x) = -(x + 1)^2 + 4 = -x^2 - 2x + 3" />

### Exercise 3

Determine the quadratic function that has roots at x = -3 and x = 2, and has a maximum value of 4.

**Answer**:

Use the factored form with p = -3 and q = 2:

<BlockMath math="f(x) = a(x - (-3))(x - 2) = a(x + 3)(x - 2)" />

Since the function has a maximum value, <InlineMath math="a < 0" />.

The vertex is at <InlineMath math="x = \frac{p + q}{2} = \frac{-3 + 2}{2} = -\frac{1}{2}" />.

Substitute x = -1/2 into the factored form and use the fact that the maximum value is 4:

<div className="flex flex-col gap-4">
  <BlockMath math="f(-\frac{1}{2}) = a(-\frac{1}{2} + 3)(-\frac{1}{2} - 2) = -a\frac{25}{4} = 4" />
  <BlockMath math="a = -\frac{16}{25}" />
</div>

Therefore, the quadratic function is:

<BlockMath math="f(x) = -\frac{16}{25}(x + 3)(x - 2) = -\frac{16}{25}x^2 - \frac{16}{25}x + \frac{96}{25}" />
