# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/quadratic-function/quadratic-function-characteristics
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/quadratic-function/quadratic-function-characteristics/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: "Characteristics of Quadratic Functions",
  description: "Learn the key characteristics of quadratic functions, including vertex, axis of symmetry, and intercepts, with clear explanations and illustrative examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/19/2025",
  subject: "Quadratic Functions",
};

## Shape of Quadratic Function Graphs

The graph of a quadratic function always forms a parabola. This parabola can open upward or downward, depending on the value of the coefficient <InlineMath math="a" />.

## Influence of Coefficient a on Graph Shape

### When a > 0

If <InlineMath math="a > 0" />, the graph of the quadratic function will open upward. This means the graph has a minimum point.

Examples of functions with <InlineMath math="a > 0" />:

- <InlineMath math="f(x) = x^2" /> (the simplest function with <InlineMath math="a = 1" />
  )
- <InlineMath math="f(x) = 2x^2 + 1" /> (example with <InlineMath math="a = 2" />
  )

<div className="mt-4">
  <LineEquation
    title={
      <>
        Quadratic Function Graph with <InlineMath math="a > 0" />
      </>
    }
    description="Graph opens upward and has a minimum point."
    cameraPosition={[5, 5, 12]}
    data={[
      {
        points: Array.from({ length: 7 }, (_, i) => {
          const x = i - 3; // x values from -3 to 3
          return { x, y: x * x, z: 0 };
        }),
        color: getColor("INDIGO"),
        labels: [
          {
            text: "f(x) = x²",
            at: 5,
            offset: [1, -1, 0],
          },
        ],
      },
    ]}
  />
</div>

### When a < 0

If <InlineMath math="a < 0" />, the graph of the quadratic function will open downward. This means the graph has a maximum point.

Examples of functions with <InlineMath math="a < 0" />:

- <InlineMath math="f(x) = -x^2" /> with <InlineMath math="a = -1" />
- <InlineMath math="f(x) = -3x^2 - 12x - 15" /> with <InlineMath math="a = -3" />

<div className="mt-4">
  <LineEquation
    title={
      <>
        Quadratic Function Graph with <InlineMath math="a < 0" />
      </>
    }
    description="Graph opens downward and has a maximum point."
    cameraPosition={[2, 2, 12]}
    data={[
      {
        points: Array.from({ length: 7 }, (_, i) => {
          const x = i - 3; // x values from -3 to 3
          return { x, y: -x * x, z: 0 };
        }),
        color: getColor("ROSE"),
        labels: [
          {
            text: "f(x) = -x²",
            at: 4,
            offset: [2, 0, 0],
          },
        ],
      },
    ]}
  />
</div>

### Why Can't a = 0?

When <InlineMath math="a = 0" />, the function form becomes <InlineMath math="f(x) = bx + c" />. This is no longer a quadratic function, but a linear function. A quadratic function must have <InlineMath math="a \neq 0" /> so that the highest power of the variable <InlineMath math="x" /> is 2.

## Important Characteristics of Quadratic Functions

### Vertex

The vertex is the highest point (if <InlineMath math="a < 0" />) or the lowest point (if <InlineMath math="a > 0" />) on the graph. The coordinates of the vertex are expressed as <InlineMath math="(-\frac{b}{2a}, f(-\frac{b}{2a}))" />.

### Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two symmetrical parts. The equation of the axis of symmetry is <InlineMath math="x = -\frac{b}{2a}" />.

### Y-Intercept

The y-intercept is obtained when <InlineMath math="x = 0" />. Its value is <InlineMath math="f(0) = c" />.

### X-Intercepts

The x-intercepts are obtained when <InlineMath math="f(x) = 0" />, i.e., when <InlineMath math="ax^2 + bx + c = 0" />. The solutions can be found using the formula:

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

## Steps to Graph a Quadratic Function

1. Determine whether the parabola opens upward (<InlineMath math="a > 0" />) or downward (<InlineMath math="a < 0" />).
2. Calculate the coordinates of the vertex <InlineMath math="(-\frac{b}{2a}, f(-\frac{b}{2a}))" />.
3. Calculate the y-intercept: <InlineMath math="(0, c)" />.
4. Calculate the x-intercepts (if any).
5. Choose several other x-values and calculate their corresponding y-values.
6. Plot all points in the coordinate system.
7. Connect the points with a parabolic curve.

## Drawing Quadratic Function Graphs

### f(x) = x² - 2x - 3

Let's graph the function <InlineMath math="f(x) = x^2 - 2x - 3" />:

1. Coefficient <InlineMath math="a = 1 > 0" />, so the parabola opens upward.
2. Vertex:

   <div className="flex flex-col gap-4">
     <BlockMath math="x = -\frac{b}{2a} = -\frac{-2}{2 \cdot 1} = 1" />
     <BlockMath math="f(1) = 1^2 - 2 \cdot 1 - 3 = 1 - 2 - 3 = -4" />
   </div>

   So the vertex is at (1, -4).

3. Y-intercept:

   <BlockMath math="f(0) = 0^2 - 2 \cdot 0 - 3 = -3" />

   So the y-intercept is at (0, -3).

4. X-intercepts: <InlineMath math="f(x) = 0" /> or <InlineMath math="x^2 - 2x - 3 = 0" />

   Using the quadratic formula:

   <div className="flex flex-col gap-4">
     <BlockMath math="x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2}" />
     <BlockMath math="x = \frac{2 + 4}{2} = 3 \text{ or } x = \frac{2 - 4}{2} = -1" />
   </div>

   So the x-intercepts are at (-1, 0) and (3, 0).

5. Let's calculate some additional points:

   <div className="flex flex-col gap-4">
     <BlockMath math="f(-2) = (-2)^2 - 2 \cdot (-2) - 3 = 4 + 4 - 3 = 5" />
     <BlockMath math="f(2) = 2^2 - 2 \cdot 2 - 3 = 4 - 4 - 3 = -3" />
   </div>

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = x^2 - 2x - 3" />
      </>
    }
    description={
      <>
        Parabola opens upward with vertex at <InlineMath math="(1, -4)" /> and
        x-intercepts at <InlineMath math="(-1, 0)" /> and{" "}
        <InlineMath math="(3, 0)" />.
      </>
    }
    cameraPosition={[2, 1, 15]}
    data={[
      {
        points: Array.from({ length: 7 }, (_, i) => {
          const x = i - 2; // x values from -2 to 4
          return { x, y: x * x - 2 * x - 3, z: 0 };
        }),
        color: getColor("TEAL"),
        labels: [
          {
            text: "f(x) = x² - 2x - 3",
            at: 5,
            offset: [3, 2, 0],
          },
          {
            text: "Vertex (1, -4)",
            at: 3,
            offset: [0, -0.5, 0],
          },
        ],
      },
    ]}
  />
</div>

### f(x) = -x²

Let's graph the function <InlineMath math="f(x) = -x^2" />:

1. Coefficient <InlineMath math="a = -1 < 0" />, so the parabola opens downward.
2. Vertex:

   <div className="flex flex-col gap-4">
     <BlockMath math="x = -\frac{b}{2a} = -\frac{0}{2 \cdot (-1)} = 0" />
     <BlockMath math="f(0) = -(0)^2 = 0" />
   </div>

   So the vertex is at (0, 0).

3. Y-intercept:

   <BlockMath math="f(0) = 0" />

   So the y-intercept is at (0, 0).

4. X-intercepts: <InlineMath math="f(x) = 0" /> or <InlineMath math="-x^2 = 0" />

   <div className="flex flex-col gap-4">
     <BlockMath math="x^2 = 0" />
     <BlockMath math="x = 0" />
   </div>

   So the x-intercept is at (0, 0).

5. Let's calculate some additional points:

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

<div className="mt-4">
  <LineEquation
    title={
      <>
        Graph of <InlineMath math="f(x) = -x²" />
      </>
    }
    description={
      <>
        Parabola opens downward with vertex at <InlineMath math="(0, 0)" />.
      </>
    }
    cameraPosition={[2, 2, 12]}
    data={[
      {
        points: Array.from({ length: 7 }, (_, i) => {
          const x = i - 3; // x values from -3 to 3
          return { x, y: -x * x, z: 0 };
        }),
        color: getColor("ORANGE"),
        labels: [
          {
            text: "f(x) = -x²",
            at: 4,
            offset: [2, 0, 0],
          },
          {
            text: "Vertex (0, 0)",
            at: 3,
            offset: [0, 0.5, 0],
          },
        ],
      },
    ]}
  />
</div>

## Table of Quadratic Function Graph Shapes

| Quadratic Function          | Graph Shape                                  |
| --------------------------- | -------------------------------------------- |
| <InlineMath math="a > 0" /> | Parabola opens upward, has a minimum point   |
| <InlineMath math="a < 0" /> | Parabola opens downward, has a maximum point |