The modern idea of a function developed over centuries. Early algebra connected formulas to unknown quantities, while analytic geometry (Descartes) linked equations to curves on coordinate axes.
Later, Euler and others formalized function notation such as \(f(x)\). Today, function thinking is central in mathematics, science, economics, and computing because it models how one quantity depends on another.
If a line passes through \((x_1,y_1)\) and \((x_2,y_2)\), slope is: \[ m=\frac{y_2-y_1}{x_2-x_1}. \] Then use \(y=mx+c\) and substitute one point to find \(c\).
Writing \(y=a(x-h)^2+k\) shows the turning point immediately: vertex \((h,k)\). This is why completing the square is useful for sketching and transformations.
Answer: \(f(5)=13\), \(f(-1)=-5\).
Answer: Domain \(x\ge 4\), range \(y\ge 0\).
Answer: No.
For a positive \(x\), there are two \(y\)-values (\(+\sqrt{x}\) and \(-\sqrt{x}\)), so one \(x\) maps to two outputs.
Answer: Shift left 1 and down 3.
Answer: \(y\)-intercept: \((0,3)\).
Answer: \(x\)-intercepts: solve \(x^2-4x+3=0\Rightarrow(x-1)(x-3)=0\), so \((1,0)\), \((3,0)\).
Answer: Vertical asymptote \(x=2\), horizontal asymptote \(y=1\).
Linear functions model plans with fixed charges plus per-unit costs, such as phone plans or transport fares.
Position-time or velocity-time graphs are function graphs. Intercepts, gradients, and curvature describe physical behavior.
Exponential, quadratic, and rational models are used to approximate trends, optimize designs, and make predictions from measured data.