Questions and Answers
Which expression is equivalent to $(2m^4)(5m^8)$? (Select one)
The function $f(x) = x$ was transformed to form $g(x) = f(x) - 23$. Which statement is true about the graphs of f and g? (Select one)
What are the x-intercepts of the function?
(-1, 0) and (3, 0)
An artist bought a set of 10 paintbrushes that contained $x$ small paintbrushes and $y$ large paintbrushes. The number of small paintbrushes in the set was 4 times the number of large paintbrushes. Which system of equations can be used to find the numbers of small paintbrushes and large paintbrushes in the set? (Select one)
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Which function could represent $f$ given $f(x) = 0$ has solutions $x = 5$ and $x = -2$? (Select one)
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Which expression is equivalent to $ abla 75$? (Select one)
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What is the solution set for $12 - 4w > 2(w - 18)$? (Select one)
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What are the factors of $2x^2 + x - 15$? Select TWO correct answers.
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Which value represents the rate of change of $y$ with respect to $x$ for this function: $2/5x - 4/7y = 3/2$? (Select one)
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A shipping company started with 40 employees. The number of employees at the shipping company increases at a rate of 7.5% each year. Which function can be used to model the number of employees $f(x)$ at the shipping company after $x$ years? (Select one)
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Which expression is a factor of $−6x^2 − 19x + 20$? (Select one)
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Which statement about $f(x) = 12x^2 - 36x + 27$ is true? (Select one)
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Which expression is equivalent to $(z + 5)(z^2 - 4z + 6)$? (Select one)
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Which expression is equivalent to $x^{-6} y^{4}/(x^{3} y)$? (Select one)
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What is the solution to $10 - 3(m - 5) = 2(5 - m)$?
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What is the slope of the line that passes through the points $(10, 18)$ and $(-5, -12)$? (Select one)
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What are the solutions to the equation $x^2 - 2 = -3x$? (Select one)
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What is the simplest form of the expression $(2x^8)(3x^{-1})^2$?
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Quadratic function $f$ has vertex (4, 15) and passes through the point (1, 20). Which equation represents $f$? (Select one)
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Which statement about the graph of the function $f(x) = 35(1.4)^x$ is true? (Select one)
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Which expression is equivalent to $k^2 - 5k - 6$? (Select one)
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Study Notes
Algebraic Expressions and Functions
- The expression (2m^4)(5m^8) simplifies to 10m^12.
- For the transformation of functions, g(x) = f(x) - 23 translates f(x) down by 23 units, confirming that f and g are parallel.
- The x-intercepts of a graph indicate points where it crosses the x-axis, specifically at (-1, 0) and (3, 0).
Systems of Equations
- A set of paintbrushes combining small (x) and large (y) brushes must satisfy both x + y = 10 and x = 4y.
Quadratic Functions
- The quadratic f(x) = x^2 with zeroes at 5 and -2 correlates to the function f(x) = 15x^2 - 11x - 14.
- Solutions to the equation f(x) = 0 can also reflect in the factorization with roots leading to expressions like (x + 3) and (2x - 5).
Inequalities and Solutions
- To solve 12 - 4w > 2(w - 18), the solution set is w < 8.
- The equation (x + 13)(x - 6) = 0 results in solutions x = -13 and x = 6.
Functions, Transformations, and Modeling
- A function representing employee growth at a shipping company can be modeled as f(x) = 40(1.075)^x due to yearly 7.5% increases.
- When analyzing a quadratic function, transformations such as shifting right by 2 units and up by 5 units are key.
Slope and Line Equations
- The slope of the line between points (10, 18) and (-5, -12) is determined as 2.
- Parallel lines maintain consistent slopes, as seen with the transformation of y = -x + 17 for line n.
Exponential Decay
- A substance's decay can be modeled by the function y = 963(0.73)^x after an initial mass of 963 grams reducing at a 27% rate per hour.
Key Points in Graph Behavior
- The function f(x) = 35(1.4)^x shows a y-intercept at (0, 35) and approaches an asymptote of y = 0 as it increases.
Evaluation of Expressions
- Simplifying expressions, e.g., ( (2x^8)(3x^{-1})^2 ), results in 18x^6 when written in the form ax^b.
- The quadratic expression k^2 - 5k - 6 factors as (k - 6)(k + 1).
Application of Functions
- The cost comparison of dance lessons at two studios leads to an equal cost solution after 5 months, illustrating practical use of linear functions.
- The vertex form of a quadratic function can help identify characteristics of its graph, such as maximum or minimum values.
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