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Magazine Chapter 14: Problem 18
Find the vertex, the focus, an equation of the axis, and an equation of the directrix of the given parabola. Draw a sketch of the graph. $$ y=3 x^{2}-3 x+3 $$
Short Answer
Expert verified
The vertex is \( (\frac{1}{2}, \frac{9}{4}) \), the focus is \( (\frac{1}{2}, \frac{7}{3}) \), the axis of symmetry is \( x = \frac{1}{2} \), and the directrix is \( y = \frac{13}{6} \).
Step by step solution
01
- Rewrite the equation in vertex form
Rewrite the quadratic equation in the form \( y = a(x-h)^2 + k \) by completing the square. Start with the given equation: \( y = 3x^2 - 3x + 3 \).Factor out the constant from the quadratic and linear terms: \( y = 3(x^2 - x) + 3 \).
02
- Complete the square
To complete the square inside the parentheses, take half of the coefficient of \( x \), square it, and add it and subtract it inside the parentheses:\( x^2 - x = x^2 - x + (\frac{-1}{2})^2 - (\frac{-1}{2})^2 = (x - \frac{1}{2})^2 - \frac{1}{4} \).The equation becomes \( y = 3((x - \frac{1}{2})^2 - \frac{1}{4}) + 3 \).
03
- Simplify the equation
Distribute the constant and simplify the equation: \( y = 3(x - \frac{1}{2})^2 - \frac{3}{4} + 3 \), which simplifies to \( y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \). So the vertex form is \( y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \).
04
- Determine the vertex
The vertex form of the quadratic equation is \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the equation \( y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \), the vertex is \( (\frac{1}{2}, \frac{9}{4}) \).
05
- Find the focus
The focus of a parabola defined by \( y = a(x - h)^2 + k \) can be found using the formula \( (h, k + \frac{1}{4a}) \). Since \( a = 3 \), the focus is \( (\frac{1}{2}, \frac{9}{4} + \frac{1}{4 \times 3}) = (\frac{1}{2}, \frac{9}{4} + \frac{1}{12}) = (\frac{1}{2}, \frac{27+1}{12}) = (\frac{1}{2}, \frac{28}{12}) = (\frac{1}{2}, \frac{7}{3}) \).
06
- Find the equation of the axis of symmetry
The axis of symmetry is a vertical line that passes through the vertex, which is \( x = \frac{1}{2} \).
07
- Find the equation of the directrix
The equation of the directrix for a parabola is \( y = k - \frac{1}{4a} \). With \( k = \frac{9}{4} \) and \( a = 3 \), the directrix is \( y = \frac{9}{4} - \frac{1}{12} = \frac{27 - 1}{12} = \frac{26}{12} = \frac{13}{6} \).
08
- Draw a sketch
To sketch the parabola, plot the vertex \( (\frac{1}{2}, \frac{9}{4}) \), the focus \( (\frac{1}{2}, \frac{7}{3}) \), and the directrix \( y = \frac{13}{6} \). The parabola opens upwards because \( a > 0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
vertex form
The vertex form of a quadratic equation is useful for identifying key properties of a parabola. It is given by the formula:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
The vertex form of the equation can be derived by completing the square on the standard form of a quadratic equation.
This transformation helps to easily identify the vertex without needing to calculate separately.
For the given equation \[ y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \], vertex form reveals the vertex is at \( ( \frac{1}{2}, \frac{9}{4} ) \).
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola.
The vertex form of the equation can be derived by completing the square on the standard form of a quadratic equation.
This transformation helps to easily identify the vertex without needing to calculate separately.
For the given equation \[ y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \], vertex form reveals the vertex is at \( ( \frac{1}{2}, \frac{9}{4} ) \).
completing the square
Completing the square is a technique used to convert a quadratic polynomial into a perfect square trinomial.
This is particularly useful for rewriting the equation of a parabola in vertex form.
Here's a quick breakdown:
In our exercise, we start from \[ y = 3x^2 - 3x + 3 \] and get to \[ y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \] by this process.
This is particularly useful for rewriting the equation of a parabola in vertex form.
Here's a quick breakdown:
- Start with the quadratic expression and factor out any coefficient of \( x^2 \).
- Take half of the linear coefficient, square it, add and subtract this square within the term.
- Factor the perfect square trinomial and simplify as needed.
In our exercise, we start from \[ y = 3x^2 - 3x + 3 \] and get to \[ y = 3(x - \frac{1}{2})^2 + \frac{9}{4} \] by this process.
focus of parabola
The focus of a parabola is a special point that lies inside the parabola, and it is used to define the parabola's geometric properties.
For the vertex form \[ y = a(x - h)^2 + k \], the focus can be found using the formula:
\[(h, k + \frac{1}{4a}) \]
This placement indicates how 'steep' or 'flat' the parabola is and helps to define its shape.
In our specific problem, with \( a = 3 \), the coordinates of the focus are computed as \[( \frac{1}{2}, \frac{9}{4} + \frac{1}{12} ) = (\frac{1}{2}, \frac{7}{3}) \].
The focus helps in graphing and understanding the parabolic motion more deeply.
For the vertex form \[ y = a(x - h)^2 + k \], the focus can be found using the formula:
\[(h, k + \frac{1}{4a}) \]
This placement indicates how 'steep' or 'flat' the parabola is and helps to define its shape.
In our specific problem, with \( a = 3 \), the coordinates of the focus are computed as \[( \frac{1}{2}, \frac{9}{4} + \frac{1}{12} ) = (\frac{1}{2}, \frac{7}{3}) \].
The focus helps in graphing and understanding the parabolic motion more deeply.
axis of symmetry
The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves.
For a parabola given by the equation in vertex form \[ y = a(x - h)^2 + k \], the axis of symmetry is the line \[( x = h) \].
This line passes through the vertex, ensuring that each point on one side of the parabola has a corresponding point on the other side.
In the given problem, since the vertex is \( (\frac{1}{2}, \frac{9}{4}) \), the axis of symmetry is \( x = \frac{1}{2} \).
This property helps in plotting the graph accurately and understanding the symmetry in the parabola.
For a parabola given by the equation in vertex form \[ y = a(x - h)^2 + k \], the axis of symmetry is the line \[( x = h) \].
This line passes through the vertex, ensuring that each point on one side of the parabola has a corresponding point on the other side.
In the given problem, since the vertex is \( (\frac{1}{2}, \frac{9}{4}) \), the axis of symmetry is \( x = \frac{1}{2} \).
This property helps in plotting the graph accurately and understanding the symmetry in the parabola.
directrix equation
The directrix of a parabola is a line perpendicular to the axis of symmetry used in the definition of the parabola.
For a parabola given by \( y = a(x - h)^2 + k \), the directrix has the equation:
\[( y = k - \frac{1}{4a}) \].
The directrix, along with the focus, helps in understanding the distances and geometric properties of the parabola.
In the provided example, with \( a = 3 \), the directrix is found as \[ y = \frac{9}{4} - \frac{1}{12} = \frac{13}{6} \].
This helps to sketch the graph accurately and visualize the parabola's orientation and shape.
For a parabola given by \( y = a(x - h)^2 + k \), the directrix has the equation:
\[( y = k - \frac{1}{4a}) \].
The directrix, along with the focus, helps in understanding the distances and geometric properties of the parabola.
In the provided example, with \( a = 3 \), the directrix is found as \[ y = \frac{9}{4} - \frac{1}{12} = \frac{13}{6} \].
This helps to sketch the graph accurately and visualize the parabola's orientation and shape.
