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Magazine Chapter 11: Problem 42
Graph each parabola. Give the vertex, axis of symmetry, domain, and range. $$ f(x)=-2(x-2)^{2}-3 $$
Short Answer
Expert verified
Vertex: (2, -3). Axis of symmetry: x=2. Domain: (-鈭�, 鈭�). Range: (-鈭�, -3].
Step by step solution
01
Identify the vertex form
The given quadratic function is already in vertex form, which is: y = a(x-h)^2 + k For the given function, y = -2(x-2)^{2}-3,where a = -2, h = 2, and k = -3.
02
Determine the vertex
Since the function is of the form y = a(x-h)^2 + k,the vertex (h, k) can be directly read off from the equation. Therefore, the vertex is (2, -3).
03
Find the axis of symmetry
The axis of symmetry can be found from the vertex form. It is the vertical line that passes through the x-coordinate of the vertex. Hence, the axis of symmetry is x = 2.
04
Determine the domain
The domain of a quadratic function is always all real numbers. Therefore, the domain is domain: (-鈭�, 鈭�).
05
Determine the range
Since the coefficient of the squared term, a = -2, is negative, the parabola opens downwards. The maximum value of the function occurs at the vertex. Thus, the range is restricted to all y-values less than or equal to the y-coordinate of the vertex.The range is: range: (-鈭�, -3].
06
Graph the parabola
To graph the parabola, plot the vertex at (2, -3). Use the axis of symmetry x = 2. Since the parabola opens downwards, draw the curve opening down from the vertex. Ensure the curvature matches the coefficient -2.
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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 function is a specific way to write the function that makes it easy to see the vertex of the parabola. It's written as:
\( y = a(x-h)^2 + k \)
Here, \(a\) dictates the width and direction of the parabola, while \(h\) and \(k\) are the coordinates of the vertex. For the function in question, \( f(x) = -2(x-2)^2 -3 \), we see that:
The vertex is \( (h, k) = (2, -3) \). Writing the quadratic function in vertex form allows easy identification of the vertex, making graphing and analyzing the function simpler.
\( y = a(x-h)^2 + k \)
Here, \(a\) dictates the width and direction of the parabola, while \(h\) and \(k\) are the coordinates of the vertex. For the function in question, \( f(x) = -2(x-2)^2 -3 \), we see that:
- \( a = -2 \) This means the parabola opens downwards.
- \(h = 2 \)
- \(k = -3 \)
The vertex is \( (h, k) = (2, -3) \). Writing the quadratic function in vertex form allows easy identification of the vertex, making graphing and analyzing the function simpler.
axis of symmetry
The axis of symmetry is a vertical line that runs through the vertex of a parabola. It essentially splits the parabola into two mirror-image halves.
For the function \( y = -2(x-2)^2 -3 \), the vertex is at \( (2, -3) \).
The axis of symmetry is the vertical line that passes through \(x = 2\).
In general, for a function in the form \( y = a(x-h)^2 + k \), the axis of symmetry is always \( x = h \).
For any point \( (x, y) \) on the parabola, there is a corresponding point \( (2h-x, y) \) on the other side of the axis of symmetry.
For the function \( y = -2(x-2)^2 -3 \), the vertex is at \( (2, -3) \).
The axis of symmetry is the vertical line that passes through \(x = 2\).
In general, for a function in the form \( y = a(x-h)^2 + k \), the axis of symmetry is always \( x = h \).
For any point \( (x, y) \) on the parabola, there is a corresponding point \( (2h-x, y) \) on the other side of the axis of symmetry.
domain and range
The domain of a quadratic function is concerned with all the possible x-values that you can input into the function.
For any quadratic function, the domain is all real numbers, written as \( (-\infty, \infty) \).
The range, however, depends on the direction the parabola opens (up or down) and the y-coordinate of the vertex. For the function \( y = -2(x-2)^2 -3 \):
Therefore, the range is all y-values less than or equal to -3, written as \( (-\infty, -3] \). Understanding domain and range helps in identifying the spread and limits of the function.
For any quadratic function, the domain is all real numbers, written as \( (-\infty, \infty) \).
The range, however, depends on the direction the parabola opens (up or down) and the y-coordinate of the vertex. For the function \( y = -2(x-2)^2 -3 \):
- The parabola opens downward (since \(a = -2\).
- The highest point (maximum value) is the y-value of the vertex, \( y = -3\).
Therefore, the range is all y-values less than or equal to -3, written as \( (-\infty, -3] \). Understanding domain and range helps in identifying the spread and limits of the function.
graphing parabolas
Graphing a parabola involves several steps. Start by plotting the vertex, then use the axis of symmetry and direction (upwards or downwards opening) to draw the curve.
For the function \( y = -2(x-2)^2 -3 \):
Using these steps, you can accurately graph any parabola, giving you a visual representation of the quadratic function. This visual can help in understanding the behavior and properties of the function.
For the function \( y = -2(x-2)^2 -3 \):
- Plot the vertex \( (2, -3) \).
- Draw the axis of symmetry line \( x = 2 \).
- Since \( a = -2 \), the parabola opens downward.
- Draw the parabola, ensuring the curvature reflects the value of \( a = -2\). The steeper the value of \(a\), the narrower the parabola.
Using these steps, you can accurately graph any parabola, giving you a visual representation of the quadratic function. This visual can help in understanding the behavior and properties of the function.
