1. **State the problem:** We want to graph the quadratic function $$f(x) = x^2 - 5x + 6$$ including its roots, vertex, and additional points for accuracy. 2. **Formula and important rules:** The quadratic function is in standard form $$f(x) = ax^2 + bx + c$$ where $$a=1$$, $$b=-5$$, and $$c=6$$. - The roots (x-intercepts) are found by factoring or using the quadratic formula. - The vertex formula is $$x = -\frac{b}{2a}$$. - The parabola opens upwards if $$a > 0$$. 3. **Find the roots:** Factor the quadratic: $$x^2 - 5x + 6 = (x - 2)(x - 3)$$ So the roots are $$x=2$$ and $$x=3$$. 4. **Find the vertex:** Calculate $$x$$ coordinate of vertex: $$x = -\frac{b}{2a} = -\frac{-5}{2 \times 1} = \frac{5}{2} = 2.5$$ Calculate $$y$$ coordinate: $$f(2.5) = (2.5)^2 - 5(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25$$ Vertex is at $$(2.5, -0.25)$$. 5. **Additional points for accuracy:** Calculate $$f(1)$$: $$f(1) = 1 - 5 + 6 = 2$$ Calculate $$f(4)$$: $$f(4) = 16 - 20 + 6 = 2$$ Points: $$(1, 2)$$ and $$(4, 2)$$. 6. **Graph description:** - Draw x-axis and y-axis with appropriate scales. - Plot roots at $$(2, 0)$$ and $$(3, 0)$$. - Plot vertex at $$(2.5, -0.25)$$. - Plot points $$(1, 2)$$ and $$(4, 2)$$. - Draw a smooth parabola opening upwards passing through these points. - Label the curve as $$f(x) = x^2 - 5x + 6$$. - Optionally, draw the axis of symmetry as a dashed line at $$x=2.5$$. Final answer: The graph of $$f(x) = x^2 - 5x + 6$$ has roots at 2 and 3, vertex at $$(2.5, -0.25)$$, and passes through points $$(1, 2)$$ and $$(4, 2)$$, forming an upward-opening parabola symmetric about $$x=2.5$$.