Questions: Consider the function: f(x) = x^2 - 4x + 3

Consider the function: f(x) = x^2 - 4x + 3
Transcript text: Consider the function: $\quad f(x)=x^{2}-4 x+3$
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Solution

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Solution Steps

To solve the given problem, we need to analyze the quadratic function \( f(x) = x^2 - 4x + 3 \).

  1. Vertex: The vertex of a quadratic function in the form \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Substitute this \( x \) value back into the function to find the \( y \)-coordinate of the vertex.

  2. Axis of Symmetry: The axis of symmetry for a quadratic function is the vertical line that passes through the vertex, given by \( x = -\frac{b}{2a} \).

  3. \( y \)-intercept: The \( y \)-intercept is the value of the function when \( x = 0 \).

Step 1: Vertex

The vertex of the function \( f(x) = x^2 - 4x + 3 \) is calculated as follows: \[ x_{\text{vertex}} = -\frac{b}{2a} = -\frac{-4}{2 \cdot 1} = 2.0 \] Substituting \( x = 2.0 \) back into the function gives: \[ y_{\text{vertex}} = f(2.0) = 2.0^2 - 4 \cdot 2.0 + 3 = -1.0 \] Thus, the vertex is \( (2.0, -1.0) \).

Step 2: Axis of Symmetry

The axis of symmetry for the quadratic function is given by the line: \[ x = 2.0 \]

Step 3: \( y \)-Intercept

The \( y \)-intercept is found by evaluating the function at \( x = 0 \): \[ y_{\text{intercept}} = f(0) = 0^2 - 4 \cdot 0 + 3 = 3 \]

Final Answer

  • Vertex: \( (2.0, -1.0) \)
  • Axis of Symmetry: \( x = 2.0 \)
  • \( y \)-Intercept: \( 3 \)

Thus, the answers are: \[ \boxed{\text{Vertex: } (2.0, -1.0)} \] \[ \boxed{\text{Axis of Symmetry: } x = 2.0} \] \[ \boxed{\text{y-Intercept: } 3} \]

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