Make a table of (x, y) pairs that will help you graph the equation. Make sure to include some negative numbers for x and some numbers that are not integers.
Plot the graph on the coordinate axes. You may need to find a few more points to plot to make the graph look smooth.﻿﻿﻿The coordinate plane provided is too small to show the whole graph. What do you think the graph looks like when x is between 0 and =? Try some values of x to test your idea.
What is the largest value that y can ever be?

To help you graph the equation, we'll first need the specific equation you're working with. Since it wasn't provided, I'll demonstrate the general process using an example equation: \( y = x^2 - 2x - 3 \). You can adapt these steps to your particular equation.

### Step 1: Choose \( x \) Values
We'll select some negative numbers, some positive numbers, and some non-integers to create a good spread of points.

| \( x \)  | \( y = x^2 - 2x - 3 \) |
|:--------:|:----------------------:|
| -2       | 1                      |
| -1       | -4                     |
| 0        | -3                     |
| 0.5      | -3.75                  |
| 1        | -4                     |
| 1.5      | -3.75                  |
| 2        | -3                     |
| 3        | 0                      |
| 4        | 5                      |

### Step 2: Plot the Points
These are the points you'll plot on the coordinate plane: 
\( (-2, 1), (-1, -4), (0, -3), (0.5, -3.75), (1, -4), (1.5, -3.75), (2, -3), (3, 0), (4, 5) \).

- You'll see that these points will form a parabola, opening upwards.
- You may need to add a few more points in the positive and negative \( x \)-range to smooth out the curve.

### Step 3: Extrapolating Beyond the Small Coordinate Plane
If the plane is too small to show the entire graph, you can infer the graph's general shape based on the equation type. Since this is a quadratic equation \( y = x^2 - 2x - 3 \), it will form a parabola that opens upward. As \( x \) approaches larger positive or negative values, the \( y \)-values will continue increasing toward infinity. You can test additional values of \( x \) to see this behavior.

### Step 4: Testing Values Between \( 0 \) and \( 2 \)
Let's test a few values of \( x \) between \( 0 \) and \( 2 \) to confirm the shape of the graph:

| \( x \)  | \( y = x^2 - 2x - 3 \) |
|:--------:|:----------------------:|
| 0.25     | -3.9375                |
| 0.75     | -3.9375                |
| 1.25     | -3.9375                |

These values help confirm the symmetry of the parabola.

### Step 5: Largest Value for \( y \)
For the quadratic equation \( y = x^2 - 2x - 3 \), the largest value \( y \) can ever be is **infinity** because as \( x \) increases in both the positive and negative direction, the parabola continues upward indefinitely.

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Do you want more details or have any questions?

### Related Questions:
1. How do you find the vertex of a parabola?
2. What is the axis of symmetry for this quadratic function?
3. How do you identify whether a parabola opens upward or downward?
4. How can you determine the x-intercepts of the equation?
5. How can the discriminant of a quadratic equation help describe the graph?

### Tip:
The vertex of a parabola gives you the maximum or minimum point, depending on the direction of the parabola (upward or downward).

Make a table of (x, y) pairs that will help you graph the equation. Make sure to include some negative numbers for x and some numbers that are not integers. Plot the graph on the coordinate axes. You may need to find a few more points to plot to make the graph look smooth.﻿﻿﻿The coordinate plane provided is too small to show the whole graph. What do you think the graph looks like when x is between 0 and =? Try some values of x to test your idea. What is the largest value that y can ever be?

Learn how to graph quadratic equations by plotting (x, y) pairs, including negative and non-integer values. Explore how the equation forms a parabola and determine the largest value for y. This solution uses the quadratic equation y = x^2 - 2x - 3 as an example, providing a step-by-step graphing guide for students in Grades 8-10.

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