Q. Quadratic function y-intercept formula.

Quadratic function: y-intercept formula — step-by-step explanation

1. Definition of y-intercept

   The y-intercept of a graph is the point where the graph crosses the y-axis. On the y-axis the x-coordinate equals zero, so to find the y-intercept you evaluate the function at x = 0.

2. Start from the standard form of a quadratic

   The general (standard) form of a quadratic function is

   \[ y = ax^{2} + bx + c \]

   Here a, b, c are constants and a is nonzero for a true quadratic.

3. Evaluate at x = 0

   Substitute x = 0 into the expression to get the y-value at the y-axis:

   \[ y(0) = a\cdot 0^{2} + b\cdot 0 + c \]

   Compute each term:

   • \(0^{2} = 0\), so \(a\cdot 0^{2} = 0\).
   • \(b\cdot 0 = 0\).
   • The remaining term is \(c\).

   Therefore

   \[ y(0) = c \]

4. Conclusion for the y-intercept

   The y-intercept point is

   \[ (0,\, c) \]

   So the y-intercept formula for a quadratic in standard form is simply the constant term c.

5. Formulas when the quadratic is given in other common forms

   If the quadratic is given in vertex form

   \[ y = a(x – h)^{2} + k \]

   then evaluate at x = 0:

   \[ y(0) = a(0 – h)^{2} + k = a h^{2} + k \]

   so the y-intercept is

   \[ (0,\, a h^{2} + k) \]

   If the quadratic is given in factored form

   \[ y = a(x – r_{1})(x – r_{2}) \]

   then evaluate at x = 0:

   \[ y(0) = a(0 – r_{1})(0 – r_{2}) = a r_{1} r_{2} \]

   so the y-intercept is

   \[ (0,\, a r_{1} r_{2}) \]

6. Special note

   If the constant term c is not explicitly given, compute it by evaluating the polynomial at x = 0 or convert the given form (vertex or factored) to standard form and read off c.

Summary: For a quadratic in standard form y = ax^{2} + bx + c the y-intercept is c and the point is (0, c). For vertex form y = a(x – h)^{2} + k the y-intercept is a h^{2} + k. For factored form y = a(x – r_{1})(x – r_{2}) the y-intercept is a r_{1} r_{2}.