A linear function is written as \( \displaystyle y = m\,x + b \). m is the slope (gradient) and b the y-intercept. Its graph is a straight line: it cuts the Y-axis at \((0,b)\) and the X-axis at \(\bigl(-\tfrac{b}{m},0\bigr)\) if \(m\neq 0\).
A quadratic function takes the form \( \displaystyle y = a\,x^{2}+b\,x+c \) with \(a\neq 0\). It draws a parabola whose vertex is at \(\bigl(-\tfrac{b}{2a},\;-\tfrac{\Delta}{4a}\bigr)\), where \(\Delta = b^{2}-4ac\) is the discriminant that determines the number of real roots: \( \Delta<0 \) → no X-intercepts; \( \Delta=0 \) → double root; \( \Delta>0 \) → two distinct roots.
Intersections of two functions are found by solving a system (or quadratic equation); the number of solutions depends on equal slopes (linear case) or on the resulting discriminant (mixed and quadratic–quadratic cases).