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 vertex is (1,-16), the intercepts are (-3,0), (5,0) and (0,-15). The axis of symmetry is \(x = 1\). The domain is all real numbers and the range is \([-16, +\infty)\]. Refer to the graph which is plotted using these identified elements.

Step 1: Find the vertex

Recall that the vertex of a quadratic function \(f(x) = ax^{2} + bx + c\) is given by \((-b/2a , f(-b/2a))\). Applying this to our function gives us: \(Vertex = (1, -16)\)

Step 2: Find the intercepts

To find the intercepts, you need to set \(f(x) = 0\) and solve for \(x\). For \(f(x) = x^{2} - 2x - 15\), the solutions are \(x = -3\) and \(x = 5\). Hence, the x-intercepts are (-3,0) and (5,0), while the y-intercept is found by setting x=0 in our function, we get the point (0,-15).

Step 3: Sketch the graph

Plot the vertex, the intercepts and a few more points if necessary. Draw a smooth curve that passes through these points in the shape of a parabola.

Step 4: Find the axis of symmetry

The equation of the axis of symmetry for the function is \(x = -b/2a\). For our function, that yields \(x = 1\).

Step 5: Determine the domain and range

For any quadratic function, the domain is all real numbers since we can plug in any number for \(x\). The range depends on whether the parabola opens upward or downward. Since the leading coefficient is positive, our parabola opens upward and the minimum point is the y-coordinate of the vertex. Therefore, the range is \([-16, +\infty)\].

Key Concepts

Vertex

The vertex of a quadratic function is like its central point or "peak." It determines where the graph curves. In the standard quadratic form, \( ax^2 + bx + c \), the vertex can be calculated using the formula \((-\frac{b}{2a}, f(-\frac{b}{2a}))\). This formula finds both the \(x\) and \(y\) coordinates of the vertex.

For example, in the function \( f(x) = x^2 - 2x - 15 \), you can find the vertex by first calculating \( -\frac{-2}{2 imes 1} \), which gives us \(1\). Then plug \(x = 1\) back into the function:

• \( f(1) = 1^2 - 2 \times 1 - 15 = -16 \)

Therefore, the vertex is at the point \((1, -16)\). The vertex is crucial because it indicates the lowest point on the graph in cases where the parabola opens upwards, such as this one.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. It essentially "splits" the parabola into two mirror-image halves. For any quadratic equation in standard form, \(ax^2 + bx + c\), the axis of symmetry can be found using the \(x\)-value of the vertex, which is \(-\frac{b}{2a}\). This creates a line expressed as \(x = \text{value}\).

In our example, the function \(f(x) = x^2 - 2x - 15\) has its vertex at \((1, -16)\), leading to an axis of symmetry given by \(x = 1\).

• This line helps ensure that for each point on the parabola to the left of the line, there is a comparable point on the right.

The axis of symmetry is an important visual guide when sketching the graph of a parabola.

Domain and Range

The domain and range of a function tell us about the set of possible \(x\) and \(y\) values, respectively. For any quadratic function, the domain is the set of all real numbers because there is always a valid \(y\) output for every \(x\) input.

However, the range is more specific and depends on the direction in which the parabola opens.

• If a parabola opens upward, as in our function \(f(x) = x^2 - 2x - 15\), it has a minimum \(y\) value at the vertex. Thus, the range consists of all real numbers greater than or equal to the \(y\)-coordinate of the vertex.
• Since our vertex is at \((1, -16)\), the range is written as \([-16, +\infty)\)

Understanding domain and range helps in determining the spread and boundaries of the graph on a coordinate grid.

Parabola

A parabola is the U-shaped curve characteristic of quadratic functions like \(f(x) = ax^2 + bx + c\). Its shape and position are primarily dictated by the coefficients \(a\), \(b\), and \(c\).

• If the coefficient \(a\) is positive, the parabola opens upwards like a smiling face. This ensures the vertex is a minimum point.
• If \(a\) is negative, the curve flips and opens downwards like a frowning face, making the vertex a maximum point."

For \(f(x) = x^2 - 2x - 15\), the parabola opens upwards. The vertex at \((1, -16)\) is the lowest point on the graph. The x-intercepts \((-3, 0)\) and \((5, 0)\) are where the parabola crosses the x-axis.
Parabolas are not just about mathematical beauty. They appear in various real-world contexts, such as defining the path of projectiles or the design of satellite dishes. Understanding their properties allows for their application across different fields.