Graph the following pair of quadratic functions and describe any similarities/differences observed in the graphs.
[tex]
\begin{array}{l}
f(x)=2 x^2 \\
g(x)=-2 x^2
\end{array}
[/tex]
a. [tex]$f$[/tex] opens downward; [tex]$g$[/tex] opens upward; both pass through different [tex]$y$[/tex]-intercepts
b. both open downward; both pass through [tex]$(0,0)$[/tex]
c. both open upward; both pass through different [tex]$y$[/tex]-intercepts
d. [tex]$f$[/tex] opens upward; [tex]$g$[/tex] opens downward; both pass through [tex]$(0,0)$[/tex]
Please select the best answer from the choices provided
Answer (2)
f ( x ) = 2 x 2 opens upward and has a y-intercept of ( 0 , 0 ) .
g ( x ) = − 2 x 2 opens downward and has a y-intercept of ( 0 , 0 ) .
f ( x ) opens upward, g ( x ) opens downward, and both pass through ( 0 , 0 ) .
The correct answer is d: f opens upward; g opens downward; both pass through ( 0 , 0 ) .
Explanation
Understanding the Problem We are given two quadratic functions, f ( x ) = 2 x 2 and g ( x ) = − 2 x 2 , and we need to describe the similarities and differences in their graphs.
Analyzing f(x) Let's analyze f ( x ) = 2 x 2 . Since the coefficient of the x 2 term is positive (2 > 0), the parabola opens upward. To find the y-intercept, we set x = 0 , so f ( 0 ) = 2 ( 0 ) 2 = 0 . Thus, the y-intercept is ( 0 , 0 ) .
Analyzing g(x) Now let's analyze g ( x ) = − 2 x 2 . Since the coefficient of the x 2 term is negative (-2 < 0), the parabola opens downward. To find the y-intercept, we set x = 0 , so g ( 0 ) = − 2 ( 0 ) 2 = 0 . Thus, the y-intercept is ( 0 , 0 ) .
Comparing the Functions Comparing the two functions, f ( x ) opens upward and g ( x ) opens downward. Both functions have the same y-intercept, which is ( 0 , 0 ) . Now we can match our observations with the given options.
Selecting the Correct Option Option a: f opens downward; g opens upward; both pass through different y -intercepts. This is incorrect because f opens upward, g opens downward, and they have the same y-intercept. Option b: both open downward; both pass through ( 0 , 0 ) . This is incorrect because f opens upward. Option c: both open upward; both pass through different y -intercepts. This is incorrect because g opens downward and they have the same y-intercept. Option d: f opens upward; g opens downward; both pass through ( 0 , 0 ) . This is correct because f opens upward, g opens downward, and both pass through ( 0 , 0 ) .
Conclusion Therefore, the correct answer is d.
Examples
Understanding quadratic functions helps in modeling projectile motion. For example, if you throw a ball, its path can be described by a quadratic function. The function's leading coefficient determines whether the parabola opens upwards or downwards, indicating the direction of motion. The y-intercept represents the initial height of the ball when thrown. By analyzing these functions, we can predict the ball's trajectory and landing point.
The functions f ( x ) = 2 x 2 and g ( x ) = − 2 x 2 have opposite orientations, with f opening upward and passing through the origin while g opens downward and also passes through the origin. Therefore, the correct answer choice is option d: f opens upward; g opens downward; both pass through ( 0 , 0 ) .
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