Graphing Functions - SAT Math

1. Identify the range of the function $f(x) = -2x^2 + 4$.

   $y \leq 4$. Parabola opens downward with maximum value $y = 4$.

2. Which function represents a parabola opening upwards?

   $y = ax^2 + bx + c$, $a > 0$. When $a > 0$, the parabola opens upward with minimum vertex.

3. What is the range of the function $y = \ln(x)$?

   All real numbers. Natural logarithm outputs all real values for positive inputs.

4. Identify the axis of symmetry for $y = ax^2 + bx + c$.

   $x = -\frac{b}{2a}$. Vertical line through the vertex of any parabola.

5. State the equation for the asymptote of $y = 2^x - 3$.

   $y = -3$. Exponential functions approach horizontal asymptotes as $x \to -\infty$.

6. What is the equation of a parabola opening upwards with vertex $(h, k)$?

   $y = a(x - h)^2 + k$. Vertex form with $a > 0$ opens upward from vertex $(h, k)$.

7. Identify the vertex form of a quadratic function.

   $y = a(x - h)^2 + k$. Shows vertex at $(h, k)$ with horizontal shifts and vertical shifts.

8. What is the general form of the exponential function?

   $y = ab^x$. Base $b$ raised to variable power $x$ with coefficient $a$.

9. Find the x-intercept of the line: $y = 3x + 6$.

   $x = -2$. Set $y = 0$ and solve: $0 = 3x + 6$, so $x = -2$.

10. State the equation of a horizontal line.

    $y = c$. Horizontal lines have zero slope and constant y-value.

11. State the domain for $f(x) = \frac{1}{x^2 - x - 6}$.

    $x \neq 3, x \neq -2$. Factor: $(x-3)(x+2) \neq 0$, so exclude both roots.

12. Identify the vertex form of a quadratic function.

    $y = a(x-h)^2 + k$. Standard vertex form with vertex at $(h,k)$ and vertical stretch $a$.

13. Find the vertex of $y = 2(x - 3)^2 + 4$.

    $(3, 4)$. Vertex form directly shows vertex coordinates $(h, k)$.

14. Identify the range of $y = 3x^2 - 5$.

    $y \geq -5$. Parabola opens upward with minimum value at vertex $y = -5$.

15. State the domain of the function $f(x) = \sqrt{x-3}$.

    $x \geq 3$. Square root requires non-negative input, so $x - 3 \geq 0$.

16. State the intercepts of $y = x^2 - 4x + 4$.

    $(2, 0)$. Perfect square trinomial $(x-2)^2$ has double root at $x = 2$.

17. Identify the transformation: $f(x) = x^2$ to $f(x) = (x + 5)^2$.

    Left 5. $(x + 5)$ represents horizontal shift left by 5 units.

18. What is the standard form of a quadratic function?

    $y = ax^2 + bx + c$. General quadratic form with leading coefficient $a$ and constant term $c$.

19. Identify the transformation: $f(x) = (x - 4)^2 + 2$.

    Right 4, Up 2. $(x - 4)$ shifts right 4, $+2$ shifts up 2.

20. Find the y-intercept of the function $f(x) = 4x^2 - 3x + 5$.

    $y = 5$. Y-intercept is the constant term when $x = 0$.

21. What is the domain of the function $f(x) = \frac{1}{x - 2}$?

    $x \neq 2$. Denominator cannot equal zero, so $x - 2 \neq 0$.

22. Which function represents exponential growth?

    $y = a \cdot b^x$, $b > 1$. Base $b > 1$ creates increasing exponential function.

23. What is the result of $f(x) = x^3$ being reflected over the y-axis?

    $f(x) = -x^3$. Reflection over y-axis changes $x$ to $-x$ in odd functions.

24. What is the general form of a linear function?

    $y = mx + b$. Slope-intercept form where $m$ is slope and $b$ is y-intercept.

25. What is the slope of the line $y = -5x + 2$?

    $m = -5$. Coefficient of $x$ in linear form $y = mx + b$.

26. What is the vertex of the quadratic function $y = x^2 - 6x + 9$?

    $(3, 0)$. Complete the square: $(x-3)^2$ gives vertex at $(3, 0)$.

27. What is the general form of a linear equation?

    $y = mx + b$. Standard linear form with slope $m$ and y-intercept $b$.

28. Find the x-intercept of the function $y = 2x + 3$.

    $x = -\frac{3}{2}$. Set $y = 0$ and solve: $0 = 2x + 3$, so $x = -\frac{3}{2}$.

29. Identify the range of the function $f(x) = |x|$.

    $y \geq 0$. Absolute value function always produces non-negative outputs.

30. What is the y-intercept of the line: $y = -4x + 7$?

    $y = 7$. Y-intercept occurs when $x = 0$, giving $y = 7$.

31. What is the shape of the graph of $y = \frac{1}{x}$?

    Hyperbola. Rational function with two branches in opposite quadrants.

32. What is the effect of $f(x) = x^2$ becoming $f(x) = -x^2$?

    Reflection over x-axis. Negative coefficient flips parabola upside down.

33. State the formula for the slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$.

    $m = \frac{y_2 - y_1}{x_2 - x_1}$. Rise over run formula between two points.

34. State the transformation: $f(x) = x^2$ to $f(x) = 3x^2$.

    Vertical stretch by 3. Coefficient greater than 1 stretches graph vertically.

35. Find the x-intercepts of $y = x^2 - 9$.

    $x = 3, x = -3$. Factor as $(x-3)(x+3) = 0$ to find roots.

36. Find the domain of the function $f(x) = \sqrt{x - 4}$.

    $x \geq 4$. Square root requires non-negative argument: $x - 4 \geq 0$.

37. Identify the equation for a circle centered at the origin.

    $x^2 + y^2 = r^2$. Circle equation with center $(0,0)$ and radius $r$.

38. What is the slope of a vertical line?

    Undefined. Vertical lines have infinite slope, expressed as undefined.

39. Determine the y-intercept of the function $y = -3x + 7$.

    $y = 7$. When $x = 0$, $y = -3(0) + 7 = 7$.

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