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Magazine Chapter 8: Problem 19
\(f(x)=-\left(x+\frac{5}{2}\right)^{2}+\frac{3}{2}\)
Short Answer
Expert verified
Vertex: \((-\frac{5}{2}, \frac{3}{2})\), Y-intercept: \((0, -\frac{19}{4})\), Opens downward.
Step by step solution
01
Identify the Function Type
The function given is in the form of \[-\left(x + \frac{5}{2}\right)^2 + \frac{3}{2}\], which is a quadratic function in vertex form \(f(x) = a(x - h)^2 + k\). Here, \(a = -1\), \(h = -\frac{5}{2}\), \(k = \frac{3}{2}\). Given the negative sign in front of the squared term, the parabola opens downwards.
02
Determine the Vertex
The vertex form of a quadratic function provides the vertex \((h, k)\) directly. From our function, \(h = -\frac{5}{2}\) and \(k = \frac{3}{2}\), so the vertex is \(\left(-\frac{5}{2}, \frac{3}{2}\right)\), which is the highest point on the graph due to the negative coefficient \(a\).
03
Assess Symmetry and Intercept
Since this is a quadratic function, it is symmetric about the line \(x = h\) or \(x = -\frac{5}{2}\). To find the y-intercept, evaluate at \(x = 0\):\[f(0) = -\left(0 + \frac{5}{2}\right)^2 + \frac{3}{2} = -\left(\frac{25}{4}\right) + \frac{3}{2} = -\frac{25}{4} + \frac{6}{4} = -\frac{19}{4}\]Thus, the y-intercept is \(\left(0, -\frac{19}{4}\right)\).
04
Graph Behavior Analysis
The parabola opens downwards with its vertex at \(-\frac{5}{2}, \frac{3}{2}\). It has no x-intercepts, as\[-(x + \frac{5}{2})^2 + \frac{3}{2} = 0 \]solving gives a non-real solution \(x = -\frac{5}{2} \pm i\sqrt{\frac{3}{2}}\). The parabola crosses the y-axis at \((0, -\frac{19}{4})\).
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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 quite useful as it provides immediate insights into the graph's shape.
This form is written as \(f(x) = a(x - h)^2 + k\). Here, \(a\) indicates the direction in which the parabola opens, while \(h\) and \(k\) correspond to the vertex coordinates \((h, k)\).
The vertex is a crucial point that either represents the maximum or minimum of the function, depending on whether the parabola opens upwards or downwards.
This form is written as \(f(x) = a(x - h)^2 + k\). Here, \(a\) indicates the direction in which the parabola opens, while \(h\) and \(k\) correspond to the vertex coordinates \((h, k)\).
The vertex is a crucial point that either represents the maximum or minimum of the function, depending on whether the parabola opens upwards or downwards.
- In our example, the vertex form is given by \(-\left(x + \frac{5}{2}\right)^2 + \frac{3}{2}\).
- This indicates that the vertex is at \((-\frac{5}{2}, \frac{3}{2})\).
- The coefficient \(a = -1\) tells us more about the parabola's direction.
Parabola Opens Downwards
The sign of \(a\) in the vertex form plays a key role in determining the orientation of the parabola.
When \(a < 0\), like in our example where \(a = -1\), the parabola opens downwards. This means that the function will have a maximum point at the vertex.
When \(a < 0\), like in our example where \(a = -1\), the parabola opens downwards. This means that the function will have a maximum point at the vertex.
- If \(a\) were positive, the parabola would instead open upwards, having a minimum point at the vertex.
- A downward opening parabola indicates the function decreases as you move away from the vertex horizontally, creating a sort of hill shape.
Graph Symmetry
The symmetry of a quadratic function is a fundamental characteristic that simplifies its analysis.
For any quadratic in vertex form, symmetry occurs about the vertical line \(x = h\). This line passes through the vertex.
Symmetry means that the left side of the parabola is a mirror image of the right side.
For any quadratic in vertex form, symmetry occurs about the vertical line \(x = h\). This line passes through the vertex.
Symmetry means that the left side of the parabola is a mirror image of the right side.
- Unlike other functions, quadratics revolve around this axis of symmetry making predictions and calculations easier.
- In our function \(-\left(x + \frac{5}{2}\right)^2 + \frac{3}{2}\), the axis of symmetry is \(x = -\frac{5}{2}\).
Y-Intercept Calculation
Calculating the y-intercept for any function helps determine where the graph crosses the y-axis.
For polynomial functions like quadratic functions, this is done by simply setting \(x = 0\) and solving for \(f(x)\).
In our example, substituting \(x = 0\) into \(-\left(x + \frac{5}{2}\right)^2 + \frac{3}{2}\) gives:
The y-intercept provides a concrete point where the graph intersects the y-axis, often serving as a starting point for sketching graphs.
For polynomial functions like quadratic functions, this is done by simply setting \(x = 0\) and solving for \(f(x)\).
In our example, substituting \(x = 0\) into \(-\left(x + \frac{5}{2}\right)^2 + \frac{3}{2}\) gives:
- \(f(0) = -\left(\frac{5}{2}\right)^2 + \frac{3}{2}\)
- \( = -\frac{25}{4} + \frac{3}{2} = -\frac{19}{4}\)
The y-intercept provides a concrete point where the graph intersects the y-axis, often serving as a starting point for sketching graphs.
