Problem 51 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function. Sketch the parabola. Find the intervals on which the function is increasing and decreasing, and find the range. $$f(x)=-x^{2}+10 x-8$$

Step-by-Step Solution

Answer

The vertex is \((-5, -83)\), the axis of symmetry is \(x = -5\). The function is increasing on the interval \((-5, +∞)\) and decreasing on the interval \((-∞, -5)\). The range of the function is all real numbers \(\leq -83\).

Step 1: Find the Vertex

The vertex of a quadratic function is given by the point \((h, k)\) where \(h = {-b \over 2a}\) and \(k = f(h)\). For this function, \(a = -1, b = 10\). So, \(h = {-(-10) \over 2(-1)} = -5\). Substitute \(h\) into \(f(x)\) to find \(k\): \(k = f(-5) = -(-5)^{2} + 10(-5) - 8 = -25 - 50 - 8 = -83\). So the vertex of the function is \((-5, -83)\).

Step 2: Find the Axis of Symmetry

The axis of symmetry of a quadratic function is the line \(x = h\). From the previous step, we know \(h = -5\), so the axis of symmetry is \(x = -5\).

Step 3: Sketch the Parabola

Draw the parabola with vertex at \((-5, -83)\) and axis of symmetry \(x=-5\). The parabola opens downwards because the coefficient of \(x^2\) is negative.

Step 4: Determine the Intervals of Increase and Decrease

Given that the parabola opens downwards, it is decreasing on the interval \((-∞, -5)\), which means to the left of \(x = -5\) and it is increasing on the interval \((-5, +∞)\), to the right of \(x = -5\).

Step 5: Find the Range

The range of a quadratic function in standard form with a negative leading coefficient is \([-∞, k]\), implying it includes negative infinity up to the vertex. From step 1, we found \(k = -83\). So, the range is all real numbers \(\leq -83\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a critical point. It represents the maximum or minimum point of the quadratic function, based on the direction the parabola opens.
The vertex is a point \(h, k\), where \(h = \frac{-b}{2a}\) and \(k = f(h)\). This formula comes from the standard form of a quadratic function \(f(x) = ax^2 + bx + c\).
Using the given quadratic function \(f(x) = -x^2 + 10x - 8\), we find the vertex by:

• Identifying \(a = -1\) and \(b = 10\)
• Calculating \(h = \frac{-10}{2(-1)} = 5\)
• Finding \(k = f(5) = -5^2 + 10(5) - 8 = -25 + 50 - 8 = 17\)

Thus, the vertex of the parabola is \(5, 17\), indicating it's the maximum point since the parabola opens downwards.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. This line splits the parabola into two mirror-image halves.
For any quadratic function, the axis of symmetry is given by the equation \(x = h\). Here, \(h\) is the x-coordinate of the vertex.
In our problem, since the vertex is \((5, 17)\), the axis of symmetry is defined as \(x = 5\).
This means that if we drew the parabola on a graph, the line \(x = 5\) would perfectly bisect the curve, ensuring each side is a reflection of the other.

Increasing and Decreasing Intervals

These intervals relate to how the values of the quadratic function change across different x-values.
The function can both increase and decrease, creating distinct segments when graphing.

• A function is increasing on an interval if the value of the function becomes larger as x increases
• It's decreasing on an interval if the value of the function becomes smaller as x increases

For the quadratic \(f(x) = -x^2 + 10x - 8\), which is a downward-opening parabola:

• It decreases in the interval \((-∞, 5)\), as the parabola falls to the vertex
• It increases in the interval \( (5, +∞) \), rising away from the vertex on the other side

Range of a Function

The range of a quadratic function describes all possible output values (y-values) produced by it.
For quadratic functions that open downwards, like \(f(x) = -x^2 + 10x - 8\), the maximum value is at the vertex.
This suggests that the range is bound from this maximum down to minus infinity.

• We previously calculated the vertex as \((5, 17)\)
• Thus, the range of the function is \([-∞, 17]\)

This means \(f(x)\) can never produce a result higher than \(17\), covering every real number below or equal to this value.