Let's analyze each quadratic equation to determine whether it has a minimum or maximum, find the Axis of Symmetry, Vertex, Domain, and Range.

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General Form of a Quadratic Equation

The standard form is $y = ax^2 + bx + c$.

• If $a > 0$, the parabola opens upwards (has a minimum).
• If $a < 0$, the parabola opens downwards (has a maximum).
• Axis of Symmetry: $x = -\frac{b}{2a}$
• Vertex: Substitute $x = -\frac{b}{2a}$ into the equation to find $y$-coordinate.
• Domain: For all quadratics, the domain is $(-\infty, \infty)$.
• Range: Determined by the vertex and the parabola's direction.

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(1) $y = 0.25x^2 - 2x + 5$

• $a = 0.25 > 0$: Parabola opens upwards → Minimum.
• Axis of Symmetry: $x = -\frac{-2}{2(0.25)} = 4$
• Vertex: $y = 0.25(4)^2 - 2(4) + 5 = 0.25(16) - 8 + 5 = 4 - 8 + 5 = 1$
  Vertex: $(4, 1)$.
• Domain: $(-\infty, \infty)$
• Range: Since the minimum $y$-value is 1 (from the vertex),
  Range: $[1, \infty)$.

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(2) $y = x^2 + 3x + 1$

• $a = 1 > 0$: Parabola opens upwards → Minimum.
• Axis of Symmetry: $x = -\frac{3}{2(1)} = -1.5$
• Vertex: $y = (-1.5)^2 + 3(-1.5) + 1 = 2.25 - 4.5 + 1 = -1.25$
  Vertex: $(-1.5, -1.25)$.
• Domain: $(-\infty, \infty)$
• Range: Since the minimum $y$-value is $-1.25$:
  Range: $[-1.25, \infty)$.

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(3) $y = 2x^2 + 4x - 2$

• $a = 2 > 0$: Parabola opens upwards → Minimum.
• Axis of Symmetry: $x = -\frac{4}{2(2)} = -1$
• Vertex: $y = 2(-1)^2 + 4(-1) - 2 = 2(1) - 4 - 2 = -4$
  Vertex: $(-1, -4)$.
• Domain: $(-\infty, \infty)$
• Range: Since the minimum $y$-value is $-4$:
  Range: $[-4, \infty)$.

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(4) $y = x^2 - 2x + 1$

• $a = 1 > 0$: Parabola opens upwards → Minimum.
• Axis of Symmetry: $x = -\frac{-2}{2(1)} = 1$
• Vertex: $y = (1)^2 - 2(1) + 1 = 1 - 2 + 1 = 0$
  Vertex: $(1, 0)$.
• Domain: $(-\infty, \infty)$
• Range: Since the minimum $y$-value is $0$:
  Range: $[0, \infty)$.

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Final Results in Tabular Form

Equation	Min/Max	Axis of Symmetry	Vertex	Domain	Range
$y = 0.25x^2 - 2x + 5$	Minimum	$x = 4$	$(4, 1)$	$(-\infty, \infty)$	$[1, \infty)$
$y = x^2 + 3x + 1$	Minimum	$x = -1.5$	$(-1.5, -1.25)$	$(-\infty, \infty)$	$[-1.25, \infty)$
$y = 2x^2 + 4x - 2$	Minimum	$x = -1$	$(-1, -4)$	$(-\infty, \infty)$	$[-4, \infty)$
$y = x^2 - 2x + 1$	Minimum	$x = 1$	$(1, 0)$	$(-\infty, \infty)$	$[0, \infty)$

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Would you like detailed steps for any specific equation?
Here are 5 related questions to extend your understanding:

1. How do we determine if a parabola has a minimum or maximum without graphing?
2. Can you find the x-intercepts (roots) for any of these equations?
3. What would the graph of each parabola look like?
4. How does the coefficient $a$ influence the steepness of the parabola?
5. How do you calculate the range when the parabola opens downward?

Tip: Always check the sign of $a$ in $y = ax^2 + bx + c$ to determine the parabola's direction and vertex type!