If y=-8x+3 is translated upward by 3 units, then reflected across the-axis, the new equation will be

To find the new equation after translating \( y = -8x + 3 \) upward by 3 units and then reflecting it across the x-axis, we need to perform these transformations step-by-step.

### Step 1: Translate Upward by 3 Units
When a function \( y = f(x) \) is translated upward by \( c \) units, the new equation becomes \( y = f(x) + c \).

Given:
\[ y = -8x + 3 \]
Translate upward by 3 units:
\[ y = (-8x + 3) + 3 \]
\[ y = -8x + 6 \]

### Step 2: Reflect Across the x-Axis
When a function \( y = f(x) \) is reflected across the x-axis, the new equation becomes \( y = -f(x) \).

Given the translated equation:
\[ y = -8x + 6 \]
Reflect across the x-axis:
\[ y = -(-8x + 6) \]
\[ y = 8x - 6 \]

Therefore, the new equation after translating \( y = -8x + 3 \) upward by 3 units and then reflecting it across the x-axis is:
\[ y = 8x - 6 \]

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 it were reflected across the y-axis instead of the x-axis?
2. What is the effect of translating the graph downward by 5 units?
3. If the original equation were translated to the right by 2 units, what would the new equation be?
4. How does a vertical stretch by a factor of 2 affect the original equation?
5. What happens to the slope of the line when it is reflected across the y-axis?
6. If the equation \( y = -8x + 3 \) were reflected across the line \( y = x \), what would the new equation be?
7. How would you graph the transformations step-by-step on a coordinate plane?
8. What is the general approach for combining multiple transformations of a function?

**Tip:** When combining multiple transformations, perform each transformation step-by-step in the given order to ensure accuracy.

Explore how the equation y = -8x + 3 transforms after being translated upward by 3 units and reflected across the x-axis. This detailed solution covers the steps involved in transformations of linear equations, suitable for high school students studying algebra and transformations.

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