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What is the range of this quadratic function?\newliney=x2−4x+4y = x^2 - 4x + 4\newlineChoices:\newline{y∣y≥2}\left\{y \mid y \geq 2\right\}\newline{y∣y≤0}\left\{y \mid y \leq 0\right\}\newline{y∣y≥0}\left\{y \mid y \geq 0\right\}\newlineall real numbers\text{all real numbers}

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Q. What is the range of this quadratic function?\newliney=x2−4x+4y = x^2 - 4x + 4\newlineChoices:\newline{y∣y≥2}\left\{y \mid y \geq 2\right\}\newline{y∣y≤0}\left\{y \mid y \leq 0\right\}\newline{y∣y≥0}\left\{y \mid y \geq 0\right\}\newlineall real numbers\text{all real numbers}
  1. Calculate x-coordinate of vertex: Find the x-coordinate of the vertex using the formula x=−b2ax = -\frac{b}{2a}. Here, a=1a = 1 and b=−4b = -4. x=−(−4)/(2⋅1)=42=2x = -(-4)/(2\cdot 1) = \frac{4}{2} = 2.
  2. Find y-coordinate of vertex: Plug x=2x = 2 into the equation to find the y-coordinate of the vertex.y=(2)2−4×(2)+4=4−8+4=0.y = (2)^2 - 4\times(2) + 4 = 4 - 8 + 4 = 0.
  3. Determine vertex and parabola direction: The vertex is (2,0)(2, 0). Since a=1a = 1, which is positive, the parabola opens upwards.
  4. Identify range of parabola: Since the parabola opens upwards and the vertex has a y-coordinate of 00, the range is all y-values greater than or equal to 00.\newlineRange: {y∣y≥0}\{y|y \geq 0\}.

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