Math Exam 1

Algebraic expressions and evaluations

1. Writing or solving algebraic problems

• Sum: $x+y+23x\; +\; y\; +\; 23$
• Difference: $z-15z\; –\; 15$
• Combine like terms with multiplication shown: $2+r+r+s+2+t+r+t+s+t+t+t=4+3\cdot r+2\cdot s+4\cdot t2\; +\; r\; +\; r\; +\; s\; +\; 2\; +\; t\; +\; r\; +\; t\; +\; s\; +\; t\; +\; t\; +\; t\; =\; 4\; +\; 3\backslash cdot\; r\; +\; 2\backslash cdot\; s\; +\; 4\backslash cdot\; t$
• Product with given values: $m\cdot n\cdot 3p=7\cdot 4\cdot 3\cdot \frac{1}{2}=42m\backslash cdot\; n\; \backslash cdot\; 3p\; =\; 7\backslash cdot\; 4\; \backslash cdot\; 3\backslash cdot\; \backslash tfrac\{1\}\{2\}\; =\; 42$
• Quotient: ${\displaystyle \frac{x}{y}}\backslash dfrac\{x\}\{y\}$
• Power: ${15}^8$
• 2. Order of operations
• Evaluate:$84-3[5+2\times 6-4]+3\times 5^{2}84\; –\; 3[5\; +\; 2\; \backslash times\; 6\; –\; 4]\; +\; 3\; \backslash times\; 5^2$
  1. Inside brackets: $5+2\times 6-4=5+12-4=135\; +\; 2\; \backslash times\; 6\; –\; 4\; =\; 5\; +\; 12\; –\; 4\; =\; 13$
  2. Multiply: $-3\times 13=-39-3\; \backslash times\; 13\; =\; -39$
  3. Power and multiply: $5^2=255^2\; =\; 25$, then $3\times 25=753\; \backslash times\; 25\; =\; 75$
  4. Combine: $84-39+75=12084\; –\; 39\; +\; 75\; =\; 120$
• Result: $120$
• 3. Distributive property
• Expand: $3x(2x-y+ab)=6x^2-3xy+3abx$
• 4. Opposite and absolute value
• For $1515$: Opposite: $-15-15$; Absolute value: $\mathrm{\mid }15\mathrm{\mid }=15|15|\; =\; 15$
• For $-9-9$: Opposite: $99$; Absolute value: $\mathrm{\mid }-9\mathrm{\mid }=9$
• 5. Evaluate the expressions
• Sum: $9+(-4)=59\; +\; (-4)\; =\; 5$
• Subtract a negative: $12-(-6)=1812\; –\; (-6)\; =\; 18$
• Subtract: $-15-4=-19-15\; –\; 4\; =\; -19$
• Multiply negatives: $-2\times (-8)=16-2\; \backslash times\; (-8)\; =\; 16$
• Divide: $-16÷4=-4$
• 6. Factor out common factors
• Expression: $6x^3+12x^{2}y=6x^2(x+2y)$
• Graphing guidance
• 7a. Parabola $y=(x+2{)}^2-1y\; =\; (x\; +\; 2)^2\; –\; 1$
• Vertex: $(-2,-1)(-2,\; -1)$
• Axis of symmetry: $x=-2x\; =\; -2$
• Opens: Upward
• Intercepts:
  • x-intercepts: Solve $(x+2{)}^2=1\Rightarrow x=-3,-1(x+2)^2\; =\; 1\; \backslash Rightarrow\; x\; =\; -3,\; -1$
  • y-intercept: $x=0\Rightarrow y=3x=0\; \backslash Rightarrow\; y=3$
• 7b. Absolute value $y=\mathrm{\mid }x-2\mathrm{\mid }y\; =\; |x\; –\; 2|$
• Vertex: $(2,0)(2,\; 0)$
• Shape: V with slopes $+1+1$ (right) and $-1-1$ (left)
• Intercepts:
  • x-intercept: $x=2x\; =\; 2$
  • y-intercept: $x=0\Rightarrow y=2x=0\; \backslash Rightarrow\; y=2$
• 7c. Two lines on one graph
• First line: $y=3x+4y\; =\; 3x\; +\; 4$ (slope $33$, y-intercept $44$)
• Second line: The expression appears unclear. Can you confirm whether it is $y=\frac{3}{2}x-1y\; =\; \backslash tfrac\{3\}\{2\}x\; –\; 1$ or $y=\frac{3}{2}-xy\; =\; \backslash tfrac\{3\}\{2\}\; –\; x$?
• Identifying function types
• 8. Function types and properties
• $y=12x+5y\; =\; 12x\; +\; 5$: Linear, not proportional (nonzero intercept).
• $y=4xy\; =\; 4x$: Linear and proportional (direct variation with constant $44$).
• $y=x^{3}y\; =\; x^3$: Cubic, odd function, passes through origin.
• Comparing rational numbers
• 9. Insert the correct inequality
• Compare: $0.6670.667\; \backslash ,$ and ${\displaystyle \frac{8}{5}}\backslash ,\backslash dfrac\{8\}\{5\}$ →
• Compare: $-5-5\; \backslash ,$ and $-3\backslash ,-3$ →
• Compare: $0.0010.001\; \backslash ,$ and $0.01\backslash ,0.01$ → $0.001<0.01$
• Rational numbers and rounding
• 10. What is a rational number?
• Definition: A rational number is any number that can be written as ${\displaystyle \frac{p}{q}}\backslash dfrac\{p\}\{q\}$ where $p,qp,\; q$ are integers and $q\ne 0q\; \backslash neq\; 0$. Its decimal form terminates or repeats.
• 11. Approximations
• Nearest ten-thousandth: $4.560246\to 4.56024.560246\; \backslash to\; 4.5602$
• Nearest hundredth: $3.14159265\to 3.143.14159265\; \backslash to\; 3.14$
• Nearest tenth: ${\displaystyle \frac{3}{2}}=1.5\to 1.5$
• Linear modeling and age functions
• 13. Snow sculpture problem
• a. Weight after $xx$ days: $W(x)=250-3xW(x)\; =\; 250\; –\; 3x$
• b. Specific times:
  • 7 days: $W(7)=250-21=229W(7)\; =\; 250\; –\; 21\; =\; 229$
  • 31 days: $W(31)=250-93=157W(31)\; =\; 250\; –\; 93\; =\; 157$
• c. Solve for time:
  • 160 lb: $250-3x=160\Rightarrow 3x=90\Rightarrow x=30250\; –\; 3x\; =\; 160\; \backslash Rightarrow\; 3x\; =\; 90\; \backslash Rightarrow\; x\; =\; 30$
  • 150 lb: $250-3x=150\Rightarrow 3x=100\Rightarrow x={\displaystyle \frac{100}{3}}\approx 33.\bar{3}250\; –\; 3x\; =\; 150\; \backslash Rightarrow\; 3x\; =\; 100\; \backslash Rightarrow\; x\; =\; \backslash dfrac\{100\}\{3\}\; \backslash approx\; 33.\backslash overline\{3\}$
• d. Assumption: The weight decreases at a constant rate of $33$ pounds per day (linear model), with no changes in conditions and no pauses in melting.
• 14. Caleb and James age functions
• a. Caleb as a function of James: $C=C(J)=J+4C\; =\; C(J)\; =\; J\; +\; 4$
• b. James as a function of Caleb: $J=J(C)=C-4J\; =\; J(C)\; =\; C\; –\; 4$
• c. Part (a) variables: Dependent: $CC$; Independent: $JJ$
• d. Part (b) variables: Dependent: $JJ$; Independent: $C$