If the graph of -9x^2+8x+5y^2-4y=20 is stretched horizontally by a factor of 8 , the equation of the stretched graph will be?

To find the equation of the graph after it has been stretched horizontally by a factor of 8, we need to modify the \( x \)-term in the original equation accordingly. 

The given equation is:
\[ -9x^2 + 8x + 5y^2 - 4y = 20 \]

When a graph is stretched horizontally by a factor of 8, the \( x \)-term is replaced by \( \frac{x}{8} \). Hence, we substitute \( x \) with \( \frac{x}{8} \) in the original equation.

Original equation:
\[ -9x^2 + 8x + 5y^2 - 4y = 20 \]

Replace \( x \) with \( \frac{x}{8} \):
\[ -9\left(\frac{x}{8}\right)^2 + 8\left(\frac{x}{8}\right) + 5y^2 - 4y = 20 \]

Simplify the substituted terms:
\[ -9\left(\frac{x^2}{64}\right) + 8\left(\frac{x}{8}\right) + 5y^2 - 4y = 20 \]
\[ -\frac{9x^2}{64} + x + 5y^2 - 4y = 20 \]

Therefore, the equation of the graph after being stretched horizontally by a factor of 8 is:
\[ -\frac{9x^2}{64} + x + 5y^2 - 4y = 20 \]

Would you like more details or have any questions?

Here are some related questions to further explore the topic:

1. How would the equation change if the graph were stretched vertically instead of horizontally?
2. What would happen to the equation if the graph were compressed horizontally by a factor of 2?
3. How would you describe the effect of a horizontal stretch on the shape of the graph?
4. Can you find the vertex of the original graph's equation?
5. What happens to the intercepts of the graph after a horizontal stretch?
6. How would a reflection over the y-axis affect the original equation?
7. If the graph were translated up by 3 units, what would the new equation be?
8. What is the general approach for transforming equations under various stretches, compressions, and translations?

**Tip:** When transforming equations, always substitute the variable being transformed with its modified expression (e.g., \( x \) with \( \frac{x}{k} \) for horizontal stretches by a factor of \( k \)) and simplify the equation accordingly.

Explore how to horizontally stretch the quadratic equation -9x^2 + 8x + 5y^2 - 4y = 20 by a factor of 8. Learn the step-by-step process with a detailed solution and understand the impact on the graph. Suitable for advanced high school students.

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