Discover how IDNLearn.com can help you find the answers you need quickly and easily. Our platform is designed to provide quick and accurate answers to any questions you may have.

Solve for [tex]x[/tex]:

[tex]\[ 3x = 6x - 2 \][/tex]

---

Question 10 of 10:

If [tex] g(x) = 4x^2 - 16 [/tex] were shifted 7 units to the right and 3 units down, what would the new equation be?

A. [tex] h(x) = 4(x-7)^2 - 19 [/tex]
B. [tex] h(x) = 4(x+7)^2 + 19 [/tex]
C. [tex] h(x) = 4(x+9)^2 - 17 [/tex]
D. [tex] h(x) = 4(x-9)^2 - 17 [/tex]


Sagot :

To determine the new equation after shifting the function [tex]\( g(x) = 4x^2 - 16 \)[/tex] 7 units to the right and 3 units down, we follow these steps:

1. Shift 7 units to the right:
- Shifting to the right involves replacing [tex]\( x \)[/tex] with [tex]\( x - a \)[/tex] in the original function, where [tex]\( a \)[/tex] is the number of units shifted.
- Therefore, for a shift of 7 units to the right, we replace [tex]\( x \)[/tex] with [tex]\( x - 7 \)[/tex]:
[tex]\[ g(x - 7) = 4(x - 7)^2 - 16 \][/tex]

2. Expand and simplify the new equation:
- First, compute [tex]\( (x - 7)^2 \)[/tex]:
[tex]\[ (x - 7)^2 = x^2 - 14x + 49 \][/tex]
- Substituting back into the function:
[tex]\[ 4(x - 7)^2 - 16 = 4(x^2 - 14x + 49) - 16 \][/tex]
- Distribute the 4 and simplify:
[tex]\[ = 4x^2 - 56x + 196 - 16 = 4x^2 - 56x + 180 \][/tex]

3. Shift 3 units down:
- Shifting a function down involves subtracting the number of units from the entire equation.
- For a shift of 3 units down, we subtract 3 from [tex]\( 4x^2 - 56x + 180 \)[/tex]:
[tex]\[ 4x^2 - 56x + 180 - 3 = 4x^2 - 56x + 177 \][/tex]

4. Express the new function:
- The new function after both shifts (7 units to the right and 3 units down) is:
[tex]\[ h(x) = 4(x - 7)^2 - 19 \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{A. \, h(x) = 4(x - 7)^2 - 19} \][/tex]