Problem 78 - Beginning and Intermediate Algebra

Question

a) Find the vertex. b) Find the axis of symmetry. c) Determine whether there is a maximum or minimum value and find that value. $$f(x)=-x^{2}+6 x-8$$

Step-by-Step Solution

Answer

a) The vertex of the parabola is at point (3, 1). b) The axis of symmetry for the given function is \(x = 3\). c) There is a maximum value of 1.

Step 1: Find the vertex coordinates, h and k.

We use the vertex formula for \(h\): $$h = \frac{-b}{2a} = \frac{-6}{2(-1)} = 3$$ Now we find the value of \(k\): $$k = f(h) = f(3) = -(3)^2 + 6(3) - 8 = -(-1) = 1$$ Thus, the vertex of the parabola is at point (3, 1).

Step 2: Determine the axis of symmetry.

The axis of symmetry is a vertical line passing through the vertex and has an equation of the form $$x=h$$. With the found vertex coordinates from Step 1, the axis of symmetry for the given function is $$x = 3$$.

Step 3: Determine whether there is a maximum or minimum value and find that value.

Since the coefficient of the \(x^2\) term in the function is -1, which is negative, the parabola opens downwards, implying that there is a maximum value. The maximum value of the function is the y-coordinate of the vertex, which we found earlier in Step 1. Therefore, the maximum value is $$k = 1$$. In summary: a) Vertex: (3, 1) b) Axis of symmetry: $$x = 3$$ c) Maximum value: 1

Key Concepts

Vertex of a Parabola

The vertex of a parabola is one of its most important points. For the quadratic function \(f(x) = ax^2 + bx + c\), the vertex can be thought of as the "tip" of the parabola, where it turns direction.
The vertex is found using the formula for the x-coordinate: \(h = \frac{-b}{2a}\). This formula gives us the point's horizontal position.
Once the x-coordinate is determined, substitute it back into the function to find the y-coordinate, \(k = f(h)\).The vertex coordinates are crucial because:

• The vertex represents the highest or lowest point of a parabola.
• The vertex form of the quadratic equation makes graphing easier \((f(x) = a(x-h)^2 + k)\).

In the example, the vertex for the function \(f(x) = -x^2 + 6x - 8\) is located at \((3, 1)\). This means \(x = 3\) is where the parabola reaches its peak, with the height \(y = 1\).

Axis of Symmetry

The axis of symmetry is an invisible line that runs vertically through the vertex of the parabola. It splits the parabola into two mirror-image halves.
For any quadratic function \(f(x) = ax^2 + bx + c\), the axis of symmetry has the equation \(x = h\), where \(h\) is the x-coordinate of the vertex.
This line helps in understanding:

• How the parabola is balanced around the vertex.
• Predicting the symmetry in the graph just by knowing the x-coordinate of its vertex.

In our example, since the vertex is at \((3, 1)\), the axis of symmetry for the parabola \(f(x) = -x^2 + 6x - 8\) is \(x = 3\). This means the graph is mirrored perfectly along the line \(x = 3\). Simple to visualize, this characteristic is silver-lined, directly from the vertex.

Maximum and Minimum Values

Quadratic functions can have either a maximum or a minimum value, depending on their orientation.
Whether a quadratic function has a maximum or minimum depends on the coefficient of the \(x^2\) term:

• If it's positive, the parabola opens upwards and has a minimum value.
• If it's negative, the parabola opens downwards and has a maximum value.

In our problem, the function \(f(x) = -x^2 + 6x - 8\) has a negative coefficient for the \(x^2\) term, indicating that it opens downwards.
This tells us that the function has a maximum value at its vertex. The y-coordinate of the vertex gives us this maximum value. For \((3, 1)\), the maximum value is \(1\), which is the peak point the function reaches.Understanding these values can guide excellence in graph analysis and interpretations.