Quadratic functions are all about parabolas. These U-shaped curves can shift up, down, left, or right. They can also stretch or compress. Understanding these transformations helps you graph quadratics easily.

The standard form, f(x) = a(x - h)^2 + k, is key. The 'a' controls direction and width, 'h' shifts horizontally, and 'k' shifts vertically. Mastering this form lets you quickly sketch any quadratic function.

Graphing Quadratic Functions Using Transformations

Vertical shifts in quadratic functions

• Standard form of a quadratic function $f(x) = a(x - h)^2 + k$
  • $a$ determines parabola direction (opens upward if $a > 0$, downward if $a < 0$) and width (larger $|a|$ results in narrower parabola)
  • $h$ represents horizontal shift (right if $h > 0$, left if $h < 0$)
  • $k$ represents vertical shift (up if $k > 0$, down if $k < 0$)
• Graphing steps:
  1. Find vertex by substituting $x = h$ into function, vertex coordinates are $(h, k)$
  2. Determine parabola direction based on sign of $a$
  3. Apply vertical shift by moving graph up or down by $k$ units ($k > 0$ shifts up, $k < 0$ shifts down by $|k|$ units)
• Examples:
  • $f(x) = (x - 2)^2 + 3$ has vertex at $(2, 3)$ and shifts up 3 units
  • $g(x) = -(x + 1)^2 - 4$ has vertex at $(-1, -4)$ and shifts down 4 units

Horizontal shifts of quadratic graphs

• Horizontal shifts move graph left or right by $h$ units
  • $h > 0$ shifts graph right by $h$ units ($x$-coordinates increase)
  • $h < 0$ shifts graph left by $|h|$ units ($x$-coordinates decrease)
• Sketching steps:
  1. Identify vertex $(h, k)$
  2. Plot vertex on coordinate plane
  3. Determine parabola direction based on sign of $a$
  4. Shift standard parabola $y = x^2$ (parent function) horizontally by $h$ units
• Examples:
  • $f(x) = (x - 3)^2$ shifts right 3 units
  • $g(x) = (x + 2)^2$ shifts left 2 units

Stretching vs compressing quadratic functions

• Value of $a$ in standard form determines vertical stretching or compressing
  • $|a| > 1$ compresses parabola vertically (appears narrower)
  • $0 < |a| < 1$ stretches parabola vertically (appears wider)
• Illustrating effects:
  1. Identify $a$ value in quadratic function
  2. Compare $a$ to standard parabola $y = x^2$ where $a = 1$
  3. Sketch parabola by applying vertical stretching or compressing factor $a$
• Examples:
  • $f(x) = 2x^2$ compresses vertically by factor of 2
  • $g(x) = \frac{1}{3}x^2$ stretches vertically by factor of $\frac{1}{3}$

Multiple transformations of quadratics

• Graphing steps:
  1. Identify $a$, $h$, and $k$ values in standard form
  2. Determine vertex $(h, k)$
  3. Plot vertex on coordinate plane
  4. Apply horizontal shift by moving graph left or right by $h$ units
  5. Apply vertical shift by moving graph up or down by $k$ units
  6. Apply stretching or compressing factor $a$ to parabola
  7. Sketch resulting parabola
• Example: $f(x) = -2(x + 1)^2 + 3$
  1. $a = -2$, $h = -1$, $k = 3$
  2. Vertex at $(-1, 3)$
  3. Plot vertex
  4. Shift left 1 unit
  5. Shift up 3 units
  6. Compress vertically by factor of 2
  7. Sketch parabola opening downward

Equations from quadratic graphs

• Determining equation from graph:
  1. Identify vertex $(h, k)$ from graph
  2. Determine parabola direction to find sign of $a$
  3. Estimate stretching or compressing factor $a$ by comparing parabola width to standard $y = x^2$
  4. Substitute $a$, $h$, and $k$ values into standard form $f(x) = a(x - h)^2 + k$
  5. Simplify equation if necessary
• Example: Given a parabola with vertex at $(2, -3)$, opening upward, and wider than $y = x^2$
  1. $h = 2$, $k = -3$
  2. Parabola opens upward, so $a > 0$
  3. Parabola is wider than standard, so $0 < a < 1$, estimate $a = \frac{1}{2}$
  4. $f(x) = \frac{1}{2}(x - 2)^2 - 3$
  5. No simplification needed

Additional Properties of Quadratic Functions

• Domain and range:
  • The domain of a quadratic function is all real numbers
  • The range depends on the direction of opening and the vertex
    • For upward-opening parabolas: $[k, \infty)$ where $k$ is the y-coordinate of the vertex
    • For downward-opening parabolas: $(-\infty, k]$ where $k$ is the y-coordinate of the vertex
• Concavity: Determined by the sign of $a$
  • If $a > 0$, the parabola is concave up (opens upward)
  • If $a < 0$, the parabola is concave down (opens downward)
• Zeros: The x-intercepts of the parabola, where $f(x) = 0$
  • Can be found by solving the quadratic equation $ax^2 + bx + c = 0$
  • The number of zeros depends on the discriminant ($b^2 - 4ac$)