https://file.aiquickdraw.com/mathbot/file/1725381145066sgnoh1dh.jpg

The equation provided in the image is:

\[ y = x^2 - 10x + 24 \]

To find and sketch the curve, we'll analyze the equation step by step:

### 1. **Identify the type of curve:**
This is a quadratic equation in the form \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = -10 \), and \( c = 24 \). The graph of a quadratic equation is a **parabola**.

### 2. **Determine the vertex:**
The vertex of a parabola given by \( y = ax^2 + bx + c \) is at:

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

Substituting the values of \( a \) and \( b \):

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

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

\[
y = 5^2 - 10(5) + 24 = 25 - 50 + 24 = -1
\]

So, the vertex of the parabola is \( (5, -1) \).

### 3. **Find the y-intercept:**
The y-intercept occurs where \( x = 0 \):

\[
y = 0^2 - 10(0) + 24 = 24
\]

Thus, the y-intercept is \( (0, 24) \).

### 4. **Find the x-intercepts:**
To find the x-intercepts, solve the equation \( y = 0 \):

\[
x^2 - 10x + 24 = 0
\]

This quadratic can be factored as:

\[
(x - 6)(x - 4) = 0
\]

Thus, the solutions are \( x = 6 \) and \( x = 4 \).

So, the x-intercepts are \( (6, 0) \) and \( (4, 0) \).

### 5. **Sketching the parabola:**
- The parabola opens upwards because the coefficient of \( x^2 \) is positive (\( a = 1 \)).
- It passes through the vertex \( (5, -1) \), the y-intercept \( (0, 24) \), and the x-intercepts \( (6, 0) \) and \( (4, 0) \).

Would you like a more detailed explanation of any of these steps? 

### Related Questions:
1. How do you determine the axis of symmetry for a parabola?
2. What is the significance of the vertex in a quadratic equation?
3. Can a parabola open downward? If so, how would the equation change?
4. How would you calculate the distance between the x-intercepts?
5. What is the role of the discriminant in determining the nature of the roots of a quadratic equation?

### Tip:
When sketching a quadratic function, always start by finding the vertex, intercepts, and direction of the parabola's opening. This provides a solid framework for the sketch.

Please find and trace the curve for the equation y = x^2 - 10x + 24

Learn how to find and sketch the curve of the quadratic equation y = x^2 - 10x + 24 with detailed steps. This problem explores algebraic concepts like finding the vertex, intercepts, and graphing parabolas. Ideal for students in Grades 9-10.

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