Honors Algebra 2 - PA TEST Review

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Questions and Answers

What is the value of !(2) if !(#) = 3# + 4?

10 (correct)

What is the output of )(2) if )(#) = # ! + 5# - 3?

7 (correct)

If f(x) = 10, what is the corresponding value of x?

2 (correct)

What transformation is applied from *(!) = √! to -(!) = √! + 3.5?

Shifts up 3.5 units Signup and view all the answers

What is the key feature of the transformation from *(!) = |!| to -(!) = |−!− 2| + 4?

Vertical shift up 4 Signup and view all the answers

What does the function notation for a function reflected over the x-axis look like if starting from f(x)?

-f(x - 1) + 2 Signup and view all the answers

What would the function notation be for a function undergoing a vertical stretch by a factor of 3 and a vertical shift 1 unit down, starting from f(x)?

3f(x - 4) - 1 Signup and view all the answers

For the function *(!) = 2# and its transformation -(!) = 2$# - 1, which transformation does this represent?

Vertical stretch and shift down Signup and view all the answers

What is the value of $x$ in the equation $x + y = 6$?

6 - y Signup and view all the answers

In the equation $3x = 4xy - 10$, what can be solved for $x$?

x = \frac{4y - 10}{3} Signup and view all the answers

What is the formula to calculate the volume of a cylinder?

V = \pi r^2h Signup and view all the answers

Based on the given mappings, is the relation a function?

Yes, because every input has a unique output. Signup and view all the answers

What is the correct domain for the given relation?

0, 1, 2, 5 Signup and view all the answers

What is the range for the relation defined by the mapping 0 → 0, 1 → 2, 2 → 3, 5 → 4?

0, 2, 3, 4 Signup and view all the answers

What value does $f(-3)$ represent when analyzing the graph of $f(x)$?

The $y$-coordinate at $x = -3$ Signup and view all the answers

If the domain of a given function is $1, 2, 3$, what would be a valid range based on typical function properties?

1, 2, 4 Signup and view all the answers

What will happen to the y-intercept if the outputs of a function are scaled?

The y-intercept will change depending on the scaling factor. Signup and view all the answers

When reflecting a graph over the x-axis and translating it 3 units up, what effect does this have on the original function's graph?

It flips downward and shifts upwards. Signup and view all the answers

In a hanger system involving a triangle and a square, which element should you include in your equation to represent the weights accurately?

The relationship between the weights and a set variable. Signup and view all the answers

Does scaling the outputs of a function affect the x-intercepts of the graph? How do you know?

No, the scaling does not change where the graph intersects the x-axis. Signup and view all the answers

What does reflecting a cubic function over the x-axis entail?

The function's general shape will remain the same but inverted. Signup and view all the answers

What equation can be written if the weight of the triangle is represented as 'x' and the weight of the square as 'y'?

2x + 3y = 0 Signup and view all the answers

If a function is transformed by translating it 3 units up, how is the new equation expressed?

f(x) + 3 Signup and view all the answers

If the output of a function is scaled down by a factor of 2, what is the impact on the graph's overall shape?

The graph will appear flatter. Signup and view all the answers

What is the solution to the equation $1(4x + 10) = 5 - 3x$?

$x = 1$ Signup and view all the answers

Which inequality represents the solution to $1/2 a + 1/3 a = a - 1$?

$a < 2$ Signup and view all the answers

In the equation $5x - 2(3 - x) = -(4 - x)$, what is the value of $x$?

$x = 1$ Signup and view all the answers

What is the correct interval notation for the solution to the inequality $-2 < 2x - 1 < 3$?

(0, 2) Signup and view all the answers

Which inequality is represented by $1/3 a + 6 ≥ -3(a - 1)$?

$a geq 15$ Signup and view all the answers

If the solution to the equation $1/2 a + 1/3 a = a - 1$ is $a$, which value of $a$ satisfies this equation?

$a = 1$ Signup and view all the answers

What does the expression $5(7 + 3)$ evaluate to?

$60$ Signup and view all the answers

How many tickets can be bought if the total cost for tickets is $34.95x + 4.255$?

$10$ Signup and view all the answers

What is the function value of $f(0)$ based on the piecewise definition provided?

3 Signup and view all the answers

Which of the following correctly describes the domain restriction for the function piece $f(x) = 3x$?

$x < 2$ Signup and view all the answers

For which value of x does the piece $h(x) = 7$ apply?

$x ext{ is greater than } 5$ Signup and view all the answers

What is the value of $f(-4)$ using the piecewise function?

3 Signup and view all the answers

What is the value of $h(-9)$?

-3 Signup and view all the answers

Which piece of the function applies when $x = 4$?

$f(x) = 6 - 2x$ Signup and view all the answers

What condition allows for using the function $g(x) = -x + 3$?

$x ext{ is greater than or equal to } 5$ Signup and view all the answers

For $f(-3)$ based on the piecewise functions provided, what is the resulting value?

-4 Signup and view all the answers

Study Notes

Solving Equations & Inequalities

To solve equations like (4x + 10 = 5 - 3x), isolate the variable through addition or subtraction.
For inequalities, express in interval notation to indicate the range of values that satisfy the condition.
Example: For (−2 < 2x - 1 < 3), calculate solutions step by step and provide interval notation.

Variable Solving

Identify which variable to solve for in given equations, such as (x + y = 6) (solve for (x)).
Familiarize with forms like (V = r^2h) for volume of a cylinder, highlighting radius (r) and height (h).

Functions, Domain & Range

Identify whether a relation is a function by checking if each input maps to a single output.
Domain refers to possible inputs while range indicates possible outputs.
Example: For the relation (0,0), (1,2), (2,3), (5,4), determine the domain (0, 1, 2, 5) and range (0, 2, 3, 4).

Function Evaluation

To find function values, compute outputs for specific inputs, such as (f(-3)) or (f(0.5)).
Utilize function notation for different operations like (g(x) = 3x + 4).

Function Transformations

Understand transformation principles, such as shifts and reflections:
- Example: Reflecting (f(x) = \sqrt{x}) over the x-axis shifts its graph downward.
- Vertical stretches change the steepness of the graph, while horizontal shifts affect the x-values.

Piecewise Functions

Create piecewise functions that define outputs based on input intervals.
Example structure: (f(x) = \begin{cases} 3x - 7 & x \leq 2 \ 3x + 5 & x < 5 \ x + 1 & x > -3 \end{cases}).

Applications of Functions

Reflect functions over axes and adjust outputs by translating vertically or horizontally.
Analyze how transforming functions impacts the intercepts; y-intercepts change with scaling outputs, while x-intercepts remain unchanged.

Real-world Applications

Use equations to model physical scenarios, like the weight distribution in structures, and apply algebraic reasoning to solve for unknowns.

General Concepts

Master algebraic manipulation, visualization of functions, and practical applications to solve complex problems effectively.
Regularly practice problem-solving using varied function types and transformations to reinforce understanding.

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

Prepare for the upcoming PA Test in Honors Algebra 2 with this comprehensive review focused on solving equations and inequalities. Practice your skills with various problems to enhance your understanding and performance in algebraic concepts.