Quadratic functions quiz

📚 Quick Study Guide

• 🔍 Definition: A quadratic function is a polynomial function of degree 2, generally represented as $f(x) = ax^2 + bx + c$, where $a \neq 0$.
• 📈 Parabola: The graph of a quadratic function is a parabola. If $a > 0$, the parabola opens upwards; if $a < 0$, it opens downwards.
• 顶点 (Vertex): The vertex of the parabola is the point where the function reaches its minimum (if $a > 0$) or maximum (if $a < 0$) value. The x-coordinate of the vertex is given by $x = -\frac{b}{2a}$.
• 📝 Axis of Symmetry: The vertical line passing through the vertex, defined by the equation $x = -\frac{b}{2a}$. The parabola is symmetric about this line.
• 🌱 Roots/Zeros: The roots or zeros of a quadratic function are the x-values for which $f(x) = 0$. These can be found by factoring, using the quadratic formula ($x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$), or completing the square.
• 💡 Discriminant: The discriminant, $\Delta = b^2 - 4ac$, determines the number of real roots: if $\Delta > 0$, there are two distinct real roots; if $\Delta = 0$, there is one real root (a repeated root); if $\Delta < 0$, there are no real roots.
• 🍎 Standard Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Practice Quiz

1. Which of the following is a quadratic function?

   1. $f(x) = 3x + 2$
   2. $f(x) = 2x^2 - 5x + 1$
   3. $f(x) = x^3 - x^2 + x - 1$
   4. $f(x) = \frac{1}{x} + 4$

2. The graph of a quadratic function is called a:

   1. Line
   2. Circle
   3. Parabola
   4. Hyperbola

3. What is the x-coordinate of the vertex of the quadratic function $f(x) = x^2 - 4x + 3$?

   1. -2
   2. 2
   3. -4
   4. 4

4. What is the discriminant of the quadratic equation $2x^2 + 3x - 5 = 0$?

   1. -31
   2. 49
   3. -49
   4. 31

5. If the discriminant of a quadratic equation is negative, how many real roots does it have?

   1. One
   2. Two
   3. Zero
   4. Infinite

6. What is the vertex form of the quadratic function $f(x) = (x - 3)^2 + 5$?

   1. $f(x) = (x + 3)^2 + 5$
   2. $f(x) = (x - 3)^2 - 5$
   3. $f(x) = (x - 5)^2 + 3$
   4. $f(x) = (x - 3)^2 + 5$

7. Which direction does the parabola open for the quadratic function $f(x) = -x^2 + 2x - 1$?

   1. Upwards
   2. Downwards
   3. To the right
   4. To the left