Worksheet for graphing quadratic equations by plotting points.

It looks like you've uploaded an image of a worksheet titled "Solving Quadratic Equations by Graphing", which includes several quadratic equations and blank coordinate planes for graphing. However, since I can't view or access images directly, I’ll help you solve this type of problem based on the typical format of such worksheets.

Let me walk through how to solve these types of problems step-by-step, assuming the equations are similar to common examples found in such worksheets.

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🔧 General Instructions:

To solve a quadratic equation by graphing:

1. Graph the quadratic function $ y = f(x) $.
2. Find the x-intercepts (where the graph crosses the x-axis).
3. The x-intercepts are the solutions (roots) of the equation $ f(x) = 0 $.

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Let’s assume the equations on your worksheet are something like:

#### 1. $ f(x) = x^2 - 4 $
- This is a parabola opening upwards.
- Vertex at $ (0, -4) $
- To find x-intercepts: set $ x^2 - 4 = 0 $
$$
x^2 = 4 \Rightarrow x = \pm 2
$$
- So, solutions: $ x = -2 $ and $ x = 2 $
- Plot the parabola with vertex at (0, -4), passing through (-2, 0) and (2, 0)

#### 2. $ f(x) = x^2 + 4 $
- Opens upward, vertex at (0, 4)
- Set $ x^2 + 4 = 0 \Rightarrow x^2 = -4 $ → No real solutions
- Graph does not cross the x-axis
- No real solutions

#### 3. $ f(x) = x^2 - 6x + 9 $
- Factor: $ (x - 3)^2 $
- Vertex at (3, 0), touches x-axis at $ x = 3 $
- One solution: $ x = 3 $ (double root)
- Graph touches x-axis at (3, 0)

#### 4. $ f(x) = x^2 - 8x + 15 $
- Factor: $ (x - 3)(x - 5) $
- Solutions: $ x = 3 $, $ x = 5 $
- Graph crosses x-axis at those points

#### 5. $ f(x) = -x^2 + 4 $
- Opens downward, vertex at (0, 4)
- Set $ -x^2 + 4 = 0 \Rightarrow x^2 = 4 \Rightarrow x = \pm 2 $
- Solutions: $ x = -2 $, $ x = 2 $

#### 6. $ f(x) = x^2 + 2x - 8 $
- Factor: $ (x + 4)(x - 2) $
- Solutions: $ x = -4 $, $ x = 2 $

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✔ How to Fill Out the Worksheet:

For each problem:

1. Graph the function on the provided coordinate plane.
- Find vertex: $ x = -\frac{b}{2a} $, then plug in to get y
- Plot a few points around the vertex
- Draw smooth curve
2. Identify where it crosses the x-axis — those are the solutions.
3. Write down the solutions (x-values).

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Example Walkthrough (Problem 1: $ f(x) = x^2 - 4 $):

1. Graph:
- Vertex: $ (0, -4) $
- Points: $ (-2, 0), (0, -4), (2, 0) $
- Parabola opens up
2. X-intercepts: $ (-2, 0) $ and $ (2, 0) $
3. Solutions: $ x = -2 $, $ x = 2 $

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📌 Final Tips:

- If the graph doesn’t cross the x-axis → no real solutions
- If it touches once → one real solution (double root)
- If it crosses twice → two real solutions

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If you can describe or type out the actual equations from the image, I can give you the exact solutions and instructions for each one!