https://file.aiquickdraw.com/mathbot/file/1726767105759galnuhim.jpg

The equation provided in the image is:

\[
y = x^2 - 10x + 29
\]

To solve the task, we need to find the vertex and identify five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right.

### Step 1: Find the vertex of the parabola
The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = \frac{-b}{2a}
\]

In this equation, \(a = 1\), \(b = -10\), and \(c = 29\). Plugging in the values:

\[
x = \frac{-(-10)}{2(1)} = \frac{10}{2} = 5
\]

The x-coordinate of the vertex is \( x = 5 \).

Now, substitute \( x = 5 \) back into the equation to find the y-coordinate of the vertex:

\[
y = 5^2 - 10(5) + 29 = 25 - 50 + 29 = 4
\]

Thus, the vertex is \( (5, 4) \).

### Step 2: Find two points to the left and two points to the right of the vertex
Let's choose points with \(x = 3\) and \(x = 4\) to the left of the vertex, and \(x = 6\) and \(x = 7\) to the right.

- **For \(x = 3\):**
  \[
  y = 3^2 - 10(3) + 29 = 9 - 30 + 29 = 8
  \]
  So, one point is \( (3, 8) \).

- **For \(x = 4\):**
  \[
  y = 4^2 - 10(4) + 29 = 16 - 40 + 29 = 5
  \]
  So, another point is \( (4, 5) \).

- **For \(x = 6\):**
  \[
  y = 6^2 - 10(6) + 29 = 36 - 60 + 29 = 5
  \]
  So, another point is \( (6, 5) \).

- **For \(x = 7\):**
  \[
  y = 7^2 - 10(7) + 29 = 49 - 70 + 29 = 8
  \]
  So, another point is \( (7, 8) \).

### Step 3: Summary of points to plot
- Vertex: \( (5, 4) \)
- Points to the left: \( (3, 8) \) and \( (4, 5) \)
- Points to the right: \( (6, 5) \) and \( (7, 8) \)

These are the five points that can be plotted on the parabola.

Would you like further details on the graphing or explanation on how to plot it? Here are some related questions to expand your understanding:

1. How is the vertex form of a quadratic equation derived from standard form?
2. What is the significance of the axis of symmetry in a parabola?
3. How do changes in the coefficients \(a\), \(b\), and \(c\) affect the shape of the parabola?
4. Why is the vertex a key point in plotting quadratic functions?
5. Can a parabola have more than one vertex?

**Tip:** When plotting parabolas, the axis of symmetry helps ensure you pick symmetric points around the vertex for a smoother curve!

Plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Learn how to find the vertex and plot five points on the parabola defined by the equation y = x^2 - 10x + 29. This tutorial covers the vertex formula, how to choose points around the vertex, and key algebraic concepts for high school students. Suitable for Grades 9-10.

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