Transform y=-x²: Shifting 6 Units Left and 1 Unit Down

Q: Why does shifting left use (x+6) instead of (x-6)?

This is the most confusing part of transformations! Think of it this way: to get the same y-value, x needs to be 6 units smaller. So when the input is (x+6), x itself is 6 less than before, shifting the graph left.

Q: How do I remember which direction vertical shifts go?

Vertical shifts are intuitive: +1 moves up, -1 moves down. The key is that vertical shifts happen outside the function, after you calculate the function value.

Q: Do I need to keep the negative sign in front?

Yes, absolutely! The negative sign in $ y = -x^2 $ makes the parabola open downward. Transformations don't change this fundamental shape - they just move the parabola around.

Q: What if I have multiple transformations to apply?

Apply them step by step: first horizontal shifts (change what's inside parentheses), then vertical shifts (add/subtract outside). Don't try to do both at once - you'll get confused!

Quadratic Transformations with Horizontal and Vertical Shifts

Which equation represents the following function shifting 6 spaces to the left and one space down?

$y=-x^2$

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Understand the problem

Which equation represents the following function shifting 6 spaces to the left and one space down?

$y=-x^2$

Step-by-step solution

To solve this problem, we will apply transformations to the original function:

Step 1: Horizontal Shift

The function $y = -x^2$ needs to be shifted 6 units to the left. In terms of algebraic manipulation, this means taking $x$ in the function and replacing it with $(x + 6)$ . The equation becomes:

$y = -(x + 6)^2$

Step 2: Vertical Shift

Next, we shift the function 1 unit down. This involves subtracting 1 from the entire expression we obtained from the horizontal shift:

$y = -(x + 6)^2 - 1$

This captures both the horizontal and vertical shifts. Therefore, the resulting function after applying these transformations is:

$y = -(x + 6)^2 - 1$

Final Answer

$y=-(x+6)^2-1$

Key Points to Remember

Essential concepts to master this topic

Horizontal Shifts: Replace x with (x+h) to shift h units left
Vertical Shifts: Add or subtract k outside: $y = f(x) + k$
Check Direction: Left means +6 inside parentheses, down means -1 outside ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal shift direction
Don't use (x-6) for shifting left = moves function right instead! Students often forget that horizontal shifts work opposite to intuition. Always remember: left shifts use (x+h), right shifts use (x-h).

Practice Quiz

Test your knowledge with interactive questions

Find the corresponding algebraic representation of the drawing:

FAQ

Everything you need to know about this question

Why does shifting left use (x+6) instead of (x-6)?

This is the most confusing part of transformations! Think of it this way: to get the same y-value, x needs to be 6 units smaller. So when the input is (x+6), x itself is 6 less than before, shifting the graph left.

How do I remember which direction vertical shifts go?

Vertical shifts are intuitive: +1 moves up, -1 moves down. The key is that vertical shifts happen outside the function, after you calculate the function value.

Do I need to keep the negative sign in front?

Yes, absolutely! The negative sign in $y = -x^2$ makes the parabola open downward. Transformations don't change this fundamental shape - they just move the parabola around.

Can I check my answer by plugging in a point?

Great idea! Pick a simple point from the original function like (0, 0). After shifting 6 left and 1 down, this point should be at (-6, -1). Verify: $y = -((-6)+6)^2 - 1 = -0 - 1 = -1$ ✓

What if I have multiple transformations to apply?

Apply them step by step: first horizontal shifts (change what's inside parentheses), then vertical shifts (add/subtract outside). Don't try to do both at once - you'll get confused!

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