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URL: https://nakafa.com/en/subjects/mathematics/quadratic-function/quadratic-function-characteristics
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Learn the key characteristics of quadratic functions, including vertex, axis of symmetry, and intercepts, through examples.

---

## 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 $$a$$.

Visible text: The graph of a quadratic function always forms a parabola. This parabola can open upward or downward, depending on the value of the coefficient .

## Influence of the Leading Coefficient on Graph Shape

### When the Leading Coefficient Is Positive

If $$a > 0$$, the graph of the quadratic function will open upward. This means the graph has a minimum point.

Visible text: If , the graph of the quadratic function will open upward. This means the graph has a minimum point.

Examples of functions with $$a > 0$$:

Visible text: Examples of functions with :

- $$f(x) = x^2$$ (the simplest function with $$a = 1$$
  )
- $$f(x) = 2x^2 + 1$$ (example with $$a = 2$$
  )

Visible text: - (the simplest function with 
 )
- (example with 
 )

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  Visible text: Quadratic Function Graph with
- description: Graph opens upward and has a minimum point.
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### When the Leading Coefficient Is Negative

If $$a < 0$$, the graph of the quadratic function will open downward. This means the graph has a maximum point.

Visible text: If , the graph of the quadratic function will open downward. This means the graph has a maximum point.

Examples of functions with $$a < 0$$:

Visible text: Examples of functions with :

- $$f(x) = -x^2$$ with $$a = -1$$
- $$f(x) = -3x^2 - 12x - 15$$ with $$a = -3$$

Visible text: - with 
- with

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  Visible text: Quadratic Function Graph with
- description: Graph opens downward and has a maximum point.
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### Why the Leading Coefficient Cannot Be Zero

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

Visible text: When , the function form becomes . This is no longer a quadratic function, but a linear function. A quadratic function must have so that the highest power of the variable is .

## Important Characteristics of Quadratic Functions

### Vertex

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

Visible text: The vertex is the highest point (if ) or the lowest point (if ) on the graph. The coordinates of the vertex are expressed as .

### 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 $$x = -\frac{b}{2a}$$.

Visible text: The axis of symmetry is a vertical line that divides the parabola into two symmetrical parts. The equation of the axis of symmetry is .

### Vertical Intercept

The $$y$$-intercept is obtained when $$x = 0$$. Its value is $$f(0) = c$$.

Visible text: The -intercept is obtained when . Its value is .

### Horizontal Intercepts

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

Visible text: The x-intercepts are obtained when , i.e., when . The solutions can be found using the formula:

```math
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
```

## Steps to Graph a Quadratic Function

1. Determine whether the parabola opens upward ($$a > 0$$) or downward ($$a < 0$$).
2. Calculate the coordinates of the vertex $$(-\frac{b}{2a}, f(-\frac{b}{2a}))$$.
3. Calculate the $$y$$-intercept: $$(0, c)$$.
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.

Visible text: 1. Determine whether the parabola opens upward () or downward ().
2. Calculate the coordinates of the vertex .
3. Calculate the -intercept: .
4. Calculate the x-intercepts (if any).
5. Choose several other values and calculate their corresponding values.
6. Plot all points in the coordinate system.
7. Connect the points with a parabolic curve.

## Drawing Quadratic Function Graphs

### Upward-Opening Parabola Example

Let's graph the function $$f(x) = x^2 - 2x - 3$$:

Visible text: Let's graph the function :

1. Coefficient $$a = 1 > 0$$, so the parabola opens upward.
2. Vertex:

   <MathContainer>
     
   
   ```math
   x = -\frac{b}{2a} = -\frac{-2}{2 \cdot 1} = 1
   ```

     
   
   ```math
   f(1) = 1^2 - 2 \cdot 1 - 3 = 1 - 2 - 3 = -4
   ```

   </MathContainer>

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

3. $$y$$-intercept:

   
   
   ```math
   f(0) = 0^2 - 2 \cdot 0 - 3 = -3
   ```

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

4. X-intercepts: $$f(x) = 0$$ or $$x^2 - 2x - 3 = 0$$

   Using the quadratic formula:

   <MathContainer>
     
   
   ```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}
   ```

     
   
   ```math
   x = \frac{2 + 4}{2} = 3 \text{ or} x = \frac{2 - 4}{2} = -1
   ```

   </MathContainer>

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

5. Let's calculate some additional points:

   <MathContainer>
     
   
   ```math
   f(-2) = (-2)^2 - 2 \cdot (-2) - 3 = 4 + 4 - 3 = 5
   ```

     
   
   ```math
   f(2) = 2^2 - 2 \cdot 2 - 3 = 4 - 4 - 3 = -3
   ```

   </MathContainer>

Visible text: 1. Coefficient , so the parabola opens upward.
2. Vertex:

 <MathContainer>
 
 

 
 

 </MathContainer>

 So the vertex is at .

3. -intercept:

 
 

 So the -intercept is at .

4. X-intercepts: or 

 Using the quadratic formula:

 <MathContainer>
 
 

 
 

 </MathContainer>

 So the x-intercepts are at and .

5. Let's calculate some additional points:

 <MathContainer>
 
 

 
 

 </MathContainer>

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$$(3, 0)$$.
  Visible text: Parabola opens upward with vertex at and
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### Downward-Opening Parabola Example

Let's graph the function $$f(x) = -x^2$$:

Visible text: Let's graph the function :

1. Coefficient $$a = -1 < 0$$, so the parabola opens downward.
2. Vertex:

   <MathContainer>
     
   
   ```math
   x = -\frac{b}{2a} = -\frac{0}{2 \cdot (-1)} = 0
   ```

     
   
   ```math
   f(0) = -(0)^2 = 0
   ```

   </MathContainer>

   So the vertex is at $$(0, 0)$$.

3. $$y$$-intercept:

   
   
   ```math
   f(0) = 0
   ```

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

4. X-intercepts: $$f(x) = 0$$ or $$-x^2 = 0$$

   <MathContainer>
     
   
   ```math
   x^2 = 0
   ```

     
   
   ```math
   x = 0
   ```

   </MathContainer>

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

5. Let's calculate some additional points:

   <MathContainer>
     
   
   ```math
   f(-2) = -((-2)^2) = -4
   ```

     
   
   ```math
   f(-1) = -((-1)^2) = -1
   ```

     
   
   ```math
   f(1) = -(1^2) = -1
   ```

     
   
   ```math
   f(2) = -(2^2) = -4
   ```

   </MathContainer>

Visible text: 1. Coefficient , so the parabola opens downward.
2. Vertex:

 <MathContainer>
 
 

 
 

 </MathContainer>

 So the vertex is at .

3. -intercept:

 
 

 So the -intercept is at .

4. X-intercepts: or 

 <MathContainer>
 
 

 
 

 </MathContainer>

 So the -intercept is at .

5. Let's calculate some additional points:

 <MathContainer>
 
 

 
 

 
 

 
 

 </MathContainer>

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## Table of Quadratic Function Graph Shapes

| Quadratic Function          | Graph Shape                                  |
| --------------------------- | -------------------------------------------- |
| $$a > 0$$ | Parabola opens upward, has a minimum point   |
| $$a < 0$$ | Parabola opens downward, has a maximum point |

Visible text: | Quadratic Function | Graph Shape |
| --------------------------- | -------------------------------------------- |
| | Parabola opens upward, has a minimum point |
| | Parabola opens downward, has a maximum point |