Let's solve the problem step by step for the quadratic function $f(x) = x^2 - 4x + 3$.

1. Create the Graph

The equation $f(x) = x^2 - 4x + 3$ is a parabola. To graph it, we:

• Identify the vertex.
• Determine the axis of symmetry.
• Find the x- and y-intercepts.
• Plot additional points.

Simplify into Vertex Form

Complete the square for $f(x) = x^2 - 4x + 3$: $f(x) = (x^2 - 4x) + 3$ $f(x) = (x - 2)^2 - 1$ Thus, the vertex form is $f(x) = (x - 2)^2 - 1$.

2. Vertex

From the vertex form $f(x) = (x - 2)^2 - 1$:

• The vertex is at $(2, -1)$.

3. Axis of Symmetry

The axis of symmetry is the vertical line passing through the vertex. Therefore:

• The axis of symmetry is $x = 2$.

4. y-Intercept

The y-intercept occurs when $x = 0$: $f(0) = 0^2 - 4(0) + 3 = 3$

• The y-intercept is $(0, 3)$.

5. x-Intercepts

The x-intercepts occur where $f(x) = 0$: $x^2 - 4x + 3 = 0$ Factorize: $(x - 3)(x - 1) = 0$ Thus: $x = 3 \quad \text{and} \quad x = 1$

• The x-intercepts are $(3, 0)$ and $(1, 0)$.

Additional Points

For a smoother graph, calculate $f(x)$ for points around the vertex:

• $f(1) = 1^2 - 4(1) + 3 = 0$ (already known, x-intercept)
• $f(3) = 3^2 - 4(3) + 3 = 0$ (already known, x-intercept)
• $f(4) = 4^2 - 4(4) + 3 = 3$
• $f(-1) = (-1)^2 - 4(-1) + 3 = 8$

Summary of Key Points

• Vertex: $(2, -1)$
• Axis of Symmetry: $x = 2$
• y-Intercept: $(0, 3)$
• x-Intercepts: $(3, 0)$ and $(1, 0)$
• Additional points: $(4, 3)$, $(-1, 8)$

Would you like me to plot this graph for you or further elaborate on any step?

Related Questions

1. How do you complete the square to rewrite quadratic equations in vertex form?
2. What is the significance of the axis of symmetry in a parabola?
3. How can you quickly find the vertex of a quadratic function in standard form $ax^2 + bx + c$?
4. What does the discriminant tell us about the x-intercepts of a quadratic function?
5. How do transformations affect the graph of a parabola?

Tip

For quadratic functions, the vertex form $f(x) = a(x-h)^2 + k$ is very useful for graphing, as it directly provides the vertex $(h, k)$.