Problem 19 - Algebra for College Students

Question

\(f(x)=-\left(x+\frac{5}{2}\right)^{2}+\frac{3}{2}\)

Step-by-Step Solution

Answer

Vertex: \((-\frac{5}{2}, \frac{3}{2})\), Y-intercept: \((0, -\frac{19}{4})\), Opens downward.

Step 1: 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.

Step 2: 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 \(a\).

Step 3: 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 \(\left(0, -\frac{19}{4}\right)\).

Step 4: 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 \((0, -\frac{19}{4})\).

Key Concepts

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.

• 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.

The vertex provides an easy snapshot of the quadratic's peaks and troughs, making analysis more straightforward.

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.

• 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.

Understanding the direction of the parabola is essential for analyzing its real-world applications such as motion paths or structural forms.

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.

• 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 \(x = -\frac{5}{2}\).

Recognizing symmetry helps in determining other important features of the graph such as its range and domain.

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:

• \(f(0) = -\left(\frac{5}{2}\right)^2 + \frac{3}{2}\)
• \( = -\frac{25}{4} + \frac{3}{2} = -\frac{19}{4}\)

This results in the y-intercept of \((0, -\frac{19}{4})\).
The y-intercept provides a concrete point where the graph intersects the y-axis, often serving as a starting point for sketching graphs.