4-2 Skills Practice: Solving Quadratic Equations by Graphing.

It looks like you've uploaded a worksheet titled "4-2 Skills Practice: Solving Quadratic Equations by Graphing". However, I can't see the actual image or its contents directly. But based on the title and typical structure of such worksheets, I can help you understand how to solve quadratic equations by graphing and guide you through common problems found on such sheets.

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🔍 What is "Solving Quadratic Equations by Graphing"?

A quadratic equation is an equation of the form:

$$
ax^2 + bx + c = 0
$$

To solve it by graphing, you:
1. Graph the quadratic function $ y = ax^2 + bx + c $
2. Find the x-intercepts (where the graph crosses the x-axis)
3. The x-intercepts are the solutions (roots) of the equation.

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✔ Steps to Solve by Graphing:

1. Write the equation in standard form: $ y = ax^2 + bx + c $
2. Graph the parabola using key points:
- Vertex
- Axis of symmetry
- Y-intercept
- Additional points
3. Find where the graph crosses the x-axis → these are the solutions.
4. If the graph doesn’t cross the x-axis, there are no real solutions.

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📌 Example Problem (Typical on Such Sheets):

Solve by graphing:
$$
y = x^2 - 4x + 3
$$

#### Step 1: Graph the equation
We can find key features:
- Vertex: Use formula $ x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2 $
- Plug $ x = 2 $ into equation: $ y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 $
→ Vertex: $ (2, -1) $
- Y-intercept: $ x = 0 $ → $ y = 0 - 0 + 3 = 3 $
- X-intercepts: Set $ y = 0 $:
$ x^2 - 4x + 3 = 0 $
Factor: $ (x - 1)(x - 3) = 0 $ → $ x = 1 $, $ x = 3 $

So, the graph crosses the x-axis at $ x = 1 $ and $ x = 3 $

✔ Solutions: $ x = 1 $ and $ x = 3 $

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🧩 Common Types of Problems on This Worksheet:

1. Given a graph, identify the solutions (x-intercepts).
2. Given a quadratic equation, graph it and find the roots.
3. Determine number of real solutions from the graph:
- 2 x-intercepts → 2 real solutions
- 1 x-intercept → 1 real solution (vertex touches x-axis)
- No x-intercepts → no real solutions

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📈 Sample Grid Interpretation

If the worksheet has graphs like this:

```
Graph A: Parabola opening up, crossing x-axis at x = -2 and x = 1
→ Solutions: x = -2, x = 1
```

```
Graph B: Parabola opening down, vertex at (1, 1), no x-intercepts
→ No real solutions
```

```
Graph C: Parabola touching x-axis at x = 2
→ One real solution: x = 2
```

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✔ How to Answer the Questions

For each problem:
- Look at the graph
- Identify where it crosses the x-axis
- List the x-values as your answers

If you’re given an equation instead of a graph:
- Sketch the parabola (or use a calculator/table)
- Find x-intercepts

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💡 Tips:

- Use symmetry to plot points efficiently.
- Label axes and scale carefully.
- If the graph is not clear, use a table of values.

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❓Need Help with a Specific Problem?

Since I can’t view the image, please describe one of the problems or share:
- The equation
- Or a description of the graph

Then I can walk you through solving it step-by-step!

Let me know!