To determine the effect on the parabola when its equation changes from [tex]\( f(x) = 2x^2 - 5x + 3 \)[/tex] to [tex]\( f(x) = 2x^2 - 5x + 17 \)[/tex], let's analyze the difference between the two equations step by step.
1. Understanding the Components of a Parabola:
The general form of a quadratic equation is [tex]\( f(x) = ax^2 + bx + c \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants. The [tex]\( c \)[/tex] term, also known as the constant term, affects the vertical position of the parabola but does not influence its shape or orientation.
2. Identifying Changes in the Equations:
- The initial equation is: [tex]\( f(x) = 2x^2 - 5x + 3 \)[/tex]
- The changed equation is: [tex]\( f(x) = 2x^2 - 5x + 17 \)[/tex]
By comparing these two equations, we can see that the coefficients of [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] (i.e., [tex]\( 2 \)[/tex] and [tex]\( -5 \)[/tex]) remain unchanged. The only difference lies in the constant term:
- Initial constant term: 3
- Changed constant term: 17
3. Calculating the Vertical Shift:
To find out the vertical shift, we subtract the initial constant term from the changed constant term:
[tex]\( 17 - 3 = 14 \)[/tex]
This positive result indicates that the parabola is shifted upward.
4. Concluding the Shift:
The change from [tex]\( f(x) = 2x^2 - 5x + 3 \)[/tex] to [tex]\( f(x) = 2x^2 - 5x + 17 \)[/tex] results in the parabola being translated upward by 14 units.
However, let's review the answer choices given in the question:
A. The parabola is translated up 1 unit.
B. The parabola is translated up 2 units.
C. The parabola is translated down 1 unit.
D. The parabola is translated down 2 units.
None of these choices correctly match the actual shift of 14 units upward. Given the provided choices, none are correct for this specific problem unless there was a typographical error in the listing of options. The correct analysis shows an upward shift of 14 units.
