Quadratic functions (graphing, factoring) Knowledge-Based Study

Study notes

1. What is a Quadratic Function?

   A quadratic function is a second-degree polynomial function that can be written in standard form as:

   $$f(x) = ax^2 + bx + c$$

   where a, b, and c are real numbers and $a \neq 0$.

   The graph of a quadratic function is a U-shaped smooth curve known as a parabola.

   • If $a > 0$, the parabola opens upward, meaning the vertex is the minimum point.
   • If $a < 0$, the parabola opens downward, meaning the vertex is the maximum point.

   From Standard Form to Factored Form

   When a quadratic expression can be factored over the real numbers, the function can be expressed in factored form (also called intercept form):

   $$f(x) = a(x - r_1)(x - r_2)$$

   The values $r_1$ and $r_2$ are the roots or zeros of the function, corresponding to the points where the graph crosses the x-axis, known as the x-intercepts: $(r_1, 0)$ and $(r_2, 0)$.

   The Zero Product Property

   To find the x-intercepts algebraically, set $f(x) = 0$:

   $$a(x - r_1)(x - r_2) = 0$$

   Because $a \neq 0$, either $x - r_1 = 0$ or $x - r_2 = 0$, giving $x = r_1$ or $x = r_2$.

   Symmetry of a Parabola

   Every parabola is symmetric with respect to a vertical line called the axis of symmetry. Because of this symmetry, the axis of symmetry lies exactly halfway between the two x-intercepts.

   The equation of the axis of symmetry is:

   $$x = \frac{r_1 + r_2}{2}$$

   Finding the Vertex

   The vertex $(h, k)$ lies directly on the axis of symmetry, so its x-coordinate is $h = \frac{r_1 + r_2}{2}$.

   To find the y-coordinate $k$, evaluate the function at $h$:

   $$k = f(h) = f\left(\frac{r_1 + r_2}{2}\right)$$

   Finding the y-Intercept

   The y-intercept occurs where the graph crosses the y-axis (where $x = 0$). Substitute $0$ into the function:

   $$y = f(0) = a(0 - r_1)(0 - r_2) = a \cdot r_1 \cdot r_2$$

   Worked Example: Graphing via Factoring

   Consider the quadratic function:

   $$f(x) = x^2 - 4x - 5$$

   1. Factor the expression: Find two numbers that multiply to $-5$ and add to $-4$. These numbers are $-5$ and $1$.
      So, $f(x) = (x - 5)(x + 1)$.
   2. Identify the x-intercepts: Set $f(x) = 0$, giving $x = 5$ and $x = -1$. The intercepts are $(5, 0)$ and $(-1, 0)$.
   3. Determine the axis of symmetry:
      $$x = \frac{5 + (-1)}{2} = \frac{4}{2} = 2$$
   4. Compute the vertex: Substitute $x = 2$ into the function:
      $$f(2) = (2 - 5)(2 + 1) = (-3)(3) = -9$$
      The vertex is $(2, -9)$. Because $a = 1 > 0$, the parabola opens upward, and this vertex is a minimum.
   5. Find the y-intercept: Evaluate $f(0)$:
      $$f(0) = 0^2 - 4(0) - 5 = -5$$
      The y-intercept is $(0, -5)$. By symmetry, the point $(4, -5)$ is also on the graph.
   6. Sketch the graph: Plot the vertex $(2, -9)$, intercepts $(-1, 0)$, $(5, 0)$, and $(0, -5)$, then connect them with a smooth parabolic curve.

   Summary of the Graphing Procedure

   • Step 1: Factor the quadratic into $f(x) = a(x - r_1)(x - r_2)$.
   • Step 2: Plot the x-intercepts at $(r_1, 0)$ and $(r_2, 0)$.
   • Step 3: Draw the vertical line of symmetry at $x = \frac{r_1 + r_2}{2}$.
   • Step 4: Calculate and plot the vertex $(h, f(h))$.
   • Step 5: Plot the y-intercept at $(0, f(0))$ and its symmetric reflection.
   • Step 6: Draw a smooth U-shaped curve passing through these key anchor points.

   Note: If a quadratic has a repeated factor, such as $f(x) = a(x - r)^2$, the parabola touches the x-axis at a single point $(r, 0)$, which also serves as the vertex.