Q: Which of the following best describes the end behavior of $f(x) = x^2 + 1$?

As $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to \infty$.

Question 1

Which sequence of transformations will convert the graph of $y = x^2$ to the graph of $y = x^2 + 1$?

Accepted Answer

Shift the graph of $y = x^2$ up by 1 unit.

Question 2

What is the domain of the function $f(x) = x^2 + 1$?

Accepted Answer

$(-\infty, \infty)$

Question 3

What is the range of the function $f(x) = x^2 + 1$?

Accepted Answer

$[1, \infty)$

Question 4

What is the equation of the axis of symmetry for the function $f(x) = x^2 + 1$?

Accepted Answer

$x = 0$

Question 5

Suppose you want to graph $y = x^2 + 1$ starting from $y = x^2$. Which transformation should you apply first?

Accepted Answer

Vertical translation upward by 1 unit.

Question 6

If the graph of $y = x^2$ is shifted up by 1 unit, what is the new vertex of the parabola?

Accepted Answer

$(0, 1)$

Question 7

Which of the following best describes the end behavior of $f(x) = x^2 + 1$?

Accepted Answer

As $x 	o \infty$, $f(x) 	o \infty$; as $x 	o -\infty$, $f(x) 	o \infty$.

Question 8

Which real-world scenario can be modeled by $f(x) = x^2 + 1$?

Accepted Answer

The height (in meters) of a ball thrown upward from a platform 1 meter above the ground, where $x$ is time in seconds and the ball's height increases quadratically.

Question 9

If $f(x) = x^2 + 1$, what is $f(-3)$?

Accepted Answer

$10$

Question 10

Which of the following statements is true about the vertex of $f(x) = x^2 + 1$?

Accepted Answer

It is the minimum point of the graph.

Question 11

Given $f(x) = x^2 + 1$, which of the following is the $y$-intercept?

Accepted Answer

$(0, 1)$

Question 12

Which transformation would change $y = x^2 + 1$ to $y = (x - 2)^2 + 1$?

Accepted Answer

Shift the graph right by 2 units.

Question 13

If $f(x) = x^2 + 1$, for which value of $x$ does $f(x) = 10$?

Accepted Answer

$x = 3$ or $x = -3$

Question 14

Which of the following best describes the symmetry of the graph of $f(x) = x^2 + 1$?

Accepted Answer

It is symmetric about the $y$-axis.

Question 15

If the function $f(x) = x^2 + 1$ models the cost (in dollars) to produce $x$ units of a product, what is the minimum possible cost?

Accepted Answer

$1$

Transformations and Analysis of Exponential Functions

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