Test Assessing Secondary Completion (TASC) Mathematics Flashcards

24 sample cards from the Mathematics deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. Solve $x^{2}-5x+6=0$ by factoring.

   $(x-2)(x-3)=0$, so $x=2$ or $x=3$.

2. What is a function, and what does function notation $f(x)$ represent?

   A function assigns exactly one output to each input. $f(x)$ denotes the output value of the function $f$ for the input $x$.

3. What are the domain and range of a function?

   The domain is the set of all allowed input values ($x$); the range is the set of all resulting output values ($y$ or $f(x)$).

4. What is the vertical line test, and what does it determine?

   If any vertical line crosses a graph more than once, the graph is not a function. It tests whether each input has exactly one output.

5. Evaluate $f(x)=2x^{2}-3$ at $x=4$.

   $f(4)=2(4)^{2}-3=2(16)-3=32-3=29$.

6. What is the slope-intercept form of a linear function, and what does each part mean?

   $y=mx+b$, where $m$ is the slope (rate of change) and $b$ is the $y$-intercept (value of $y$ when $x=0$).

7. How do you compute the slope between two points $(x_1,y_1)$ and $(x_2,y_2)$?

   $m=\dfrac{y_2-y_1}{x_2-x_1}$, the change in $y$ over the change in $x$.

8. What is the slope of a line that goes through $(1,2)$ and $(5,10)$?

   $m=\dfrac{10-2}{5-1}=\dfrac{8}{4}=2$.

9. How can you tell if a function is linear or nonlinear from a table of values?

   A function is linear if it has a constant rate of change (equal $\Delta y$ for equal $\Delta x$); otherwise it is nonlinear.

10. What is the shape of the graph of a quadratic function, and what is its turning point called?

    The graph of $y=ax^{2}+bx+c$ is a parabola; its turning point is the vertex. It opens upward if $a>0$ and downward if $a<0$.

11. How do you find the $x$-coordinate of the vertex of $y=ax^{2}+bx+c$?

    $x=-\dfrac{b}{2a}$; substitute it back into the function to get the $y$-coordinate.

12. When building a linear function from a real-world context, what do the slope and $y$-intercept typically represent?

    The slope represents the constant rate of change (per-unit amount), and the $y$-intercept represents the starting or fixed initial value.

13. A taxi charges a $\$3$ flat fee plus $\$2$ per mile. Write a function for the cost $C$ of riding $m$ miles.

    $C(m)=2m+3$, where $3$ is the fixed fee and $2$ is the rate per mile.

14. What are complementary angles and supplementary angles?

    Complementary angles sum to $90^{\circ}$; supplementary angles sum to $180^{\circ}$.

15. What is the sum of the interior angles of any triangle, and how are triangles classified by their angles?

    The interior angles sum to $180^{\circ}$. By angle: acute (all $<90^{\circ}$), right (one $=90^{\circ}$), obtuse (one $>90^{\circ}$).

16. What does it mean for two triangles to be similar, and what is true about their sides?

    Similar triangles have equal corresponding angles and proportional corresponding sides (the same shape, possibly different size).

17. Give the formulas for the perimeter and area of a rectangle with length $l$ and width $w$.

    Perimeter $P=2l+2w$; area $A=lw$.

18. What are the formulas for the area of a triangle and the circumference and area of a circle?

    Triangle area $A=\frac{1}{2}bh$. Circle circumference $C=2\pi r$; circle area $A=\pi r^{2}$.

19. What is the area of a circle with radius $5$? Leave your answer in terms of $\pi$.

    $A=\pi r^{2}=\pi(5)^{2}=25\pi$.

20. Give the volume formulas for a rectangular prism and a cylinder.

    Rectangular prism: $V=lwh$. Cylinder: $V=\pi r^{2}h$.

21. What is the formula for the surface area of a rectangular prism with dimensions $l$, $w$, and $h$?

    $SA=2lw+2lh+2wh$, the sum of the areas of all six faces.

22. What is the volume of a sphere of radius $r$?

    $V=\frac{4}{3}\pi r^{3}$.

23. State the Pythagorean theorem and what each variable represents.

    $a^{2}+b^{2}=c^{2}$, where $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse (the side opposite the right angle).

24. A right triangle has legs of length $3$ and $4$. Find the hypotenuse.

    $c=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=\sqrt{25}=5$.

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