Problem 28 - College Algebra

Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. $$f(x)-x^{2}-2 x-15$$

Step-by-Step Solution

Answer

The sketch of the function will show a parabola opening upward, with vertex at (1, -16), the x-intercepts at (5, 0) and (-3, 0), and the y-intercept at (0,-15). The axis of symmetry is the line \(x = 1\). The domain of the function is \(-\infty, \infty\) and the range is \([-16, \infty)\)

Step 1: Identify the Coefficients and Intercept

Looking at the function \(f(x) = x^2 - 2x - 15\), we can identify the coefficients a, b, and c in the quadratic standard form \(f(x) = ax^2 + bx + c\). Here \(a = 1\), \(b = -2\), and \(c = -15\). The constant \(c = -15\) is the y-intercept.

Step 2: Calculate the Vertex and the Axis of Symmetry

To find the vertex of the parabola, one can use the formula \(-(b/2a, f(-b/2a))\). We substitute \(a = 1\) and \(b = -2\) into the formula, we find the x-coordinate of the vertex to be \(1\). Substituting \(x = 1\) into the function, we find the y-coordinate to be \(-16\). So the vertex is \((1, -16)\). The equation for the axis of symmetry is \(x = h\), where \(h\) is the x-coordinate of the vertex, thus in this case \(x = 1\).

Step 3: Find the x-intercepts

To find the x-intercepts set \(f(x) = 0\) and solve for \(x\). This gives the equation \(x^2 - 2x - 15 = 0\). Solve this equation by factoring, yields \((x - 5)(x + 3) = 0\), from which we find the roots \(x = 5\) and \(x = -3\), which are the x-intercepts.

Step 4: Determine the Domain and Range

Because the quadratic function \(f(x)\) is defined for all real numbers, the domain is \(-\infty, \infty\). Since the coefficient on \(x^2\) is positive and the vertex is the minimum point on the graph, the range is \(-16, \infty\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point that represents either the maximum or minimum value of a quadratic function. It's where the curve changes direction. In our function, \(f(x) = x^2 - 2x - 15\), the vertex is found using the formula for the vertex of \((h, k)\), where \(h = \frac{-b}{2a}\). Here, \(a = 1\) and \(b = -2\), so \(h = 1\). To find \(k\), substitute \(x = 1\) back into the function, giving \(f(1) = -16\). Thus, the vertex is \((1, -16)\).

• If \(a\) is positive, the parabola opens upwards, and the vertex is a minimum.
• If \(a\) is negative, the parabola opens downwards, making the vertex a maximum.

This vertex \((1, -16)\) is the lowest point on our graph.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola, dividing it into two symmetrical halves. For the function \(f(x) = x^2 - 2x - 15\), this line can be found using \(x = h\), where \(h\) is the x-coordinate of the vertex. Thus, our axis of symmetry is \(x = 1\).
This concept is important because:

• It helps in graphing the parabola accurately.
• It acts as a mirror line, reflecting each point from one side to the other.

By knowing the axis, you can easily find points that lie on either side of it.

Domain and Range

Understanding the domain and range of a quadratic function helps in knowing the inputs and possible outputs. The domain of a quadratic function is all real numbers, symbolized as \(-\infty, \infty\). This is because you can plug any real number into \(x\) and get a corresponding \(y\).
For the function \(f(x) = x^2 - 2x - 15\), the range is affected by the vertex. With the vertex at \((1, -16)\) and since the parabola opens upwards (as \(a = 1\) is positive),

• The vertex gives the minimum value. Thus, the range is \([-16, \infty)\).

This means \(y\) values start at \(-16\) and go to infinity.

X-Intercepts

X-intercepts are points where the graph of the quadratic crosses the x-axis. For these, \(f(x) = 0\). In the function \(f(x) = x^2 - 2x - 15\), you set the expression to zero and factor to find the roots: \((x - 5)(x + 3) = 0\).
Solving this gives:

• \(x = 5\)
• \(x = -3\)

These intercepts show where the parabola touches the x-axis, which is essential for creating a complete sketch of the graph. It also helps in understanding real-world applications, like finding when a projectile might hit the ground.