Graph the relationship $$ h=10 t-t^{2} $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's analyze and graph the quadratic relationship $$ h = 10t - t^{2} $$.

### 1. Understanding the Equation

The equation $$ h = 10t - t^{2} $$ is a quadratic equation in the form:

$$ h = -t^{2} + 10t $$

This represents a parabola that opens **downward** because the coefficient of $$ t^{2} $$ is negative.

### 2. Key Features of the Graph

#### a. **Vertex**

The vertex of a parabola $$ h = at^{2} + bt + c $$ occurs at:

$$ t = -\frac{b}{2a} $$

For $$ h = -t^{2} + 10t $$:

- $$ a = -1 $$
- $$ b = 10 $$

So,

$$ t = -\frac{10}{2(-1)} = 5 $$

To find the corresponding $$ h $$-value:

$$ h = 10(5) - (5)^2 = 50 - 25 = 25 $$

**Vertex:** $$ (5, 25) $$

#### b. **Axis of Symmetry**

The axis of symmetry is the vertical line that passes through the vertex:

$$ t = 5 $$

#### c. **Intercepts**

- **T-intercepts (where $$ h = 0 $$):**

Set $$ h = 0 $$:

$$ 0 = 10t - t^{2} $$
$$ t^{2} - 10t = 0 $$
$$ t(t - 10) = 0 $$

So, $$ t = 0 $$ and $$ t = 10 $$.

**T-intercepts:** $$ (0, 0) $$ and $$ (10, 0) $$

- **H-intercept (where $$ t = 0 $$):**

When $$ t = 0 $$, $$ h = 0 $$.

**H-intercept:** $$ (0, 0) $$ (same as one of the t-intercepts)

#### d. **Direction of Opening**

Since the coefficient of $$ t^{2} $$ is negative ($$ a = -1 $$), the parabola opens **downward**.

### 3. Table of Values

| $$ t $$ | $$ h = 10t - t^{2} $$ |
|---------|-----------------------|
| 0       | 0                     |
| 1       | 10(1) - 1 = 9         |
| 2       | 20 - 4 = 16           |
| 3       | 30 - 9 = 21           |
| 4       | 40 - 16 = 24          |
| 5       | 50 - 25 = 25          |
| 6       | 60 - 36 = 24          |
| 7       | 70 - 49 = 21          |
| 8       | 80 - 64 = 16          |
| 9       | 90 - 81 = 9           |
| 10      | 100 - 100 = 0         |

### 4. Sketching the Graph

Using the key features and the table of values, here's a step-by-step guide to sketch the graph:

1. **Plot the Vertex:** $$ (5, 25) $$.
2. **Plot the T-intercepts:** $$ (0, 0) $$ and $$ (10, 0) $$.
3. **Draw the Axis of Symmetry:** A vertical dashed line through $$ t = 5 $$.
4. **Plot Additional Points:** Use the table above to plot points like $$ (2, 16) $$, $$ (3, 21) $$, etc.
5. **Connect the Points:** Draw a smooth, downward-opening parabola passing through all the points.

### 5. Graph Representation

While I can't provide a visual graph directly, here's a rough sketch using text:

```
 h
 |
25|          *
20|        *   *
15|      *       *
10|    *           *
5 |  *               *
0 |*---------------------*----- t
   0    2   4   5   6   8  10
```

- The highest point is at $$ (5, 25) $$.
- The parabola crosses the $$ t $$-axis at $$ t = 0 $$ and $$ t = 10 $$.

### 6. Interpretation

This quadratic relationship could represent, for example, the height $$ h $$ of an object in meters at time $$ t $$ seconds if it's thrown upwards with an initial velocity, considering gravity's effect.

### 7. Summary of Key Points

- **Vertex:** $$ (5, 25) $$
- **Axis of Symmetry:** $$ t = 5 $$
- **T-intercepts:** $$ t = 0 $$ and $$ t = 10 $$
- **Opening Direction:** Downward

This comprehensive analysis should help you visualize and understand the graph of the equation $$ h = 10t - t^{2} $$.
The graph of $$ h = 10t - t^{2} $$ is a downward-opening parabola with its vertex at $$ (5, 25) $$. It crosses the t-axis at $$ t = 0 $$ and $$ t = 10 $$.

Graph the relationship

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Pre Calculus Questions