1. First, understand the function's formula. For example, if you have $f(x) = x^2 - 4x + 3$, this tells you how to calculate the output $y$ for each input $x$. 2. Identify key features of the graph such as intercepts and extrema: - The $y$-intercept is where $x = 0$. - The $x$-intercepts (or roots) are where $f(x) = 0$. - Extrema are points where the graph has a maximum or minimum. 3. Find the $y$-intercept by plugging in $x = 0$: $$f(0) = 0^2 - 4 \cdot 0 + 3 = 3.$$ So the graph crosses the $y$-axis at $(0, 3)$. 4. To find $x$-intercepts, solve $f(x) = 0$: $$x^2 - 4x + 3 = 0.$$ Factor or use the quadratic formula: $$(x - 1)(x - 3) = 0,$$ so $x = 1$ or $x = 3$. These are points where the graph crosses the $x$-axis. 5. Find extrema by finding critical points where the derivative is zero: $$f'(x) = 2x - 4.$$ Set equal to zero: $$2x - 4 = 0 \Rightarrow x = 2.$$ Evaluate $f(2)$: $$f(2) = 2^2 - 4 \cdot 2 + 3 = 4 - 8 + 3 = -1.$$ So the graph has a minimum at $(2, -1)$. 6. Plot these points and sketch the curve based on these key features. Symmetry, end behavior, and continuity also help shape the graph. This process helps you understand the graph of any function better by identifying important points and behaviors.