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Explain the transformation $y = |2x - 3| + 4$ in terms of shifts and reflections.
Explain the transformation $y = |2x - 3| + 4$ in terms of shifts and reflections.
The transformation involves a horizontal shift of 3 units to the right, a vertical shift of 4 units upwards, and a reflection in the x-axis.
Solve the equation $2 ext{cos}(x + rac{ ext{$rac{ ext{$ ext{$ ext{$rac{ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{1}}{2}$} ext{cos}(x + 1)) = 0$ for $0 extless x extless 2 ext{$ ext{$ ext{pi}$}$}.
Solve the equation $2 ext{cos}(x + rac{ ext{$rac{ ext{$ ext{$ ext{$rac{ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{$ ext{1}}{2}$} ext{cos}(x + 1)) = 0$ for $0 extless x extless 2 ext{$ ext{$ ext{pi}$}$}.
The solutions are $x = rac{ ext{$ ext{$ ext{3}}{2}$}$} ext{pi}$ and $x = rac{ ext{$ ext{$ ext{5}}{2}$}$} ext{pi}$.
Find the x-coordinate of the vertex of the quadratic function $f(x) = 3x^2 + 4x - 6$.
Find the x-coordinate of the vertex of the quadratic function $f(x) = 3x^2 + 4x - 6$.
The x-coordinate of the vertex is $-rac{4}{6} = -rac{2}{3}$.
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Study Notes
Transformation of the Absolute Value Function
- The transformation $y = |2x - 3| + 4$ can be broken down into several components:
- Horizontal shift 3 units to the right
- Horizontal scaling by a factor of 2
- Reflection over the x-axis
- Vertical shift 4 units up
Solving a Trigonometric Equation
- The equation $2\cos(x + \frac{1}{2}) = 0$ is a trigonometric equation
- To solve for $x$, we need to find the values of $x$ in the interval $0 \leq x \leq 2\pi$ that satisfy the equation
- The solution involves finding the inverse cosine of both sides of the equation
Quadratic Function Vertex
- The quadratic function $f(x) = 3x^2 + 4x - 6$ has a vertex
- To find the x-coordinate of the vertex, we can use the formula $x = -\frac{b}{2a}$
- Plugging in the values $a = 3$ and $b = 4$, we get the x-coordinate of the vertex
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