Precalculus Functions and Their Graphs Flashcards

24 sample cards from the Functions and Their Graphs deck

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

1. What is the general form of a linear function and what does each parameter represent?

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

2. What is a constant function, and what does its graph look like?

   A function of the form $f(x) = c$ for a fixed real number $c$. Its graph is a horizontal line, and its slope is $0$.

3. What is the defining property of the average/instantaneous rate of change of a linear function $f(x)=mx+b$?

   It is constant everywhere and equals the slope $m$, regardless of the interval chosen.

4. Describe the shape and domain/range of the identity function $f(x) = x$.

   A straight line through the origin with slope $1$; domain and range are both $(-\infty, \infty)$. It is odd and increasing everywhere.

5. Describe the graph of the power function $f(x) = x^{2}$ (squaring function).

   A parabola opening upward with vertex at the origin; domain $(-\infty,\infty)$, range $[0,\infty)$; even function, symmetric about the $y$-axis.

6. Describe the graph of the cubing function $f(x) = x^{3}$.

   An S-shaped curve through the origin; domain and range both $(-\infty,\infty)$; odd function, symmetric about the origin, increasing everywhere.

7. State the domain and range of the square root function $f(x) = \sqrt{x}$.

   Domain $[0, \infty)$ and range $[0, \infty)$; the graph starts at the origin and increases.

8. State the domain and range of the cube root function $f(x) = \sqrt[3]{x}$.

   Both domain and range are $(-\infty, \infty)$; it is an odd function symmetric about the origin.

9. Give the piecewise definition of the absolute value function $f(x) = |x|$.

   $$|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$$

10. Describe the graph, domain, and range of $f(x) = |x|$.

    A V-shape with vertex at the origin; domain $(-\infty,\infty)$, range $[0,\infty)$; even function, decreasing on $(-\infty,0]$ and increasing on $[0,\infty)$.

11. Define the greatest integer (floor) function $f(x) = \lfloor x \rfloor$.

    $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$. For example, $\lfloor 2.7 \rfloor = 2$ and $\lfloor -1.3 \rfloor = -2$.

12. What type of graph does the greatest integer function produce, and what is its range?

    A step function consisting of horizontal segments (steps) that jump at each integer; its range is the set of all integers $\mathbb{Z}$.

13. Write the standard (vertex) form of a quadratic function and identify the vertex.

    $f(x) = a(x-h)^{2} + k$ with $a \neq 0$; the vertex is $(h, k)$ and the axis of symmetry is $x = h$.

14. For $f(x) = ax^{2} + bx + c$, give the $x$-coordinate of the vertex.

    $$x = -\frac{b}{2a}$$, and the $y$-coordinate is $f\!\left(-\frac{b}{2a}\right)$.

15. How does the sign of $a$ affect the parabola $f(x)=ax^{2}+bx+c$?

    If $a > 0$ the parabola opens upward and the vertex is a minimum; if $a < 0$ it opens downward and the vertex is a maximum.

16. How does the graph of $y = f(x) + c$ (with $c>0$) compare to $y = f(x)$?

    It is a vertical shift upward by $c$ units. $y = f(x) - c$ shifts the graph downward by $c$ units.

17. How does the graph of $y = f(x - c)$ (with $c>0$) compare to $y = f(x)$?

    It is a horizontal shift to the right by $c$ units. $y = f(x + c)$ shifts the graph left by $c$ units.

18. Describe the transformation from $y = f(x)$ to $y = -f(x)$.

    A reflection across the $x$-axis; each output is negated so the graph flips vertically.

19. Describe the transformation from $y = f(x)$ to $y = f(-x)$.

    A reflection across the $y$-axis; each input is negated so the graph flips horizontally.

20. For $y = a\,f(x)$ with $a > 1$ versus $0 < a < 1$, describe the transformation.

    If $a > 1$ it is a vertical stretch by factor $a$; if $0 < a < 1$ it is a vertical compression (shrink) toward the $x$-axis.

21. For $y = f(bx)$ with $b > 1$ versus $0 < b < 1$, describe the horizontal transformation.

    If $b > 1$ it is a horizontal compression by factor $\frac{1}{b}$; if $0 < b < 1$ it is a horizontal stretch by factor $\frac{1}{b}$.

22. In what order should transformations generally be applied to graph $y = a\,f(b(x-h)) + k$?

    Horizontal shifts/stretches/reflections first (inside the function), then vertical stretches/reflections, and finally vertical shifts. A common order is: horizontal shift, stretch/compress, reflect, then vertical shift.

23. Given $f$ and $g$, write the definitions of $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$, and $\left(\frac{f}{g}\right)(x)$.

    $(f+g)(x)=f(x)+g(x)$, $(f-g)(x)=f(x)-g(x)$, $(fg)(x)=f(x)\,g(x)$, and $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}$ where $g(x)\neq 0$.

24. What is the domain of an arithmetic combination such as $(f+g)$ or $(fg)$, and how does the quotient $\frac{f}{g}$ differ?

    The domain is the intersection of the domains of $f$ and $g$. For $\frac{f}{g}$ you must additionally exclude all $x$ where $g(x) = 0$.

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