Practice Test 5 - Algebra 1

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. The graphs of three functions are shown on the same coordinate plane: $f(x)=2x+5$ (linear), $g(x)=x^2$ (quadratic), and $h(x)=2^x$ (exponential). Which statement best describes what happens for large positive $x$?
2. Use the factored form $P(x)=-(x-4)(x+2)(x-1)$ to identify the zeros and sketch a rough graph, showing x-intercepts and correct end behavior.
3. Solve the inequality: $3x - 5 > 16$
4. Exponent rules for integers include the power-of-a-power property $(b^a)^c = b^{ac}$. To extend this rule to rational exponents, we want $(b^{1/n})^n = b^{(1/n) \cdot n} = b^1 = b$ for $b > 0$ and integer $n > 1$. Explain why this forces the definition $b^{1/n} = \sqrt[n]{b}$.
5. A student solves the equation $x^2-5x+6=0$ and writes: 1) $x^2-5x+6=0$ 2) $(x-2)(x-3)=0$ 3) $x-2=-3$ or $x-3=-2$ 4) $x=-1$ or $x=1$ Identify the error in the student work shown.​
6. Find all intersection points of the line and the parabola by solving the system: $\begin{cases} y = -x + 2 \\ y = x^2 \end{cases}$​
7. If $g(x)=x^2+3x-4$, what is the value of $g(-2)$?
8. Determine whether the function $g(x)=2x^2-3$ has a constant rate of change. Use the idea of $\Delta y/\Delta x$ over equal intervals (no calculus).
9. A medication amount in the body is modeled by $M(t)=100(0.97)^t$, where $t$ is in hours. What is the percent decay rate per hour?
10. Write the recursive formula for the geometric sequence $6,\ 18,\ 54,\ 162,\ \dots$.
11. Complete the square to solve $x^2 - 4x + 5 = 0$ for complex solutions. Write answers in $a \pm bi$ form, using $i^2 = -1$.
12. Solve using the elimination method: $\begin{cases} 2x + 3y = 12 \\ 2x - y = 4 \end{cases}$ What is the solution $(x, y)$?
13. Let $f(x)=x^2+2$ and $g(x)=x-1$. Find $(f\cdot g)(x)$.
14. Let $f(x)=\sqrt{x}$ and $g(x)=\dfrac{1}{x}$. Find the domain of $(f+g)(x)$.​
15. Which statement correctly explains why the definition $27^{2/3}=(\sqrt[3]{27})^2$ is consistent with extending integer exponent properties?
16. An exponential function has the values shown in the table. Write the function in the form $f(x)=a\cdot b^x$. $x: 0, 1, 2, 3$ $f(x): 6, 12, 24, 48$
17. A movie theater charges a $9 ticket plus $2 for each snack you buy. Write an explicit formula for the total cost $C(s)$ (in dollars) in terms of the number of snacks $s$.
18. Fill in the reasoning: A student solves $7-2x=15$. 1) $7-2x=15$ 2) $-2x=8$ What justification is used in going from line 1 to line 2?
19. A purchase total on a receipt is $47.23891 before rounding. For paying in dollars and cents (to the nearest $0.01), which reporting shows appropriate precision?
20. A rectangular garden has a length that is 5 feet more than its width. The area is 84 square feet. Write and solve an equation to find the width of the garden. Let $w$ = the width (in feet).

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