Problem 27 - Algebra and Trigonometry

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. $$ f(x)=x^{2}-2 x-3 $$

Step-by-Step Solution

Answer

The vertex of the parabola is at the point (1, -4), the intercepts are at (3, 0) and (-1, 0) for x, and (0, -3) for y. Its axis of symmetry is x = 1. The domain of the function is (-\(\infty\), \(\infty\)) and its range is [-4, \(\infty\)).

Step 1: Finding the Vertex of the Parabola

The vertex is given by the point (h, k), where \( h = -B/2A \) and \( A = 1 \), \( B = -2 \) from the general quadratic formula \( f(x) = Ax^2 + Bx + C \). Thus, \( h = -(-2)/(2*1) = 1 \). To find k, plug h into the original quadratic equation: \( f(h) = (1)^2 - 2(1) - 3 = -4 \). Therefore, the vertex is (1, -4).

Step 2: Finding the Intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve the given quadratic equation: \( x^2 - 2x - 3 = 0 \). Factoring the quadratic equation, we get: \( (x - 3)(x + 1) = 0 \). The solutions to the equation are \( x = 3 \) and \( x = -1 \). Therefore, the x-intercepts are (3, 0) and (-1, 0). The y-intercept can be found by evaluating the function at \( x = 0 \): \( f(0) = (0)^2 - 2(0) - 3 = -3 \). Therefore, the y-intercept is (0, -3).

Step 3: Finding the Axis of Symmetry

The axis of symmetry of the parabola is given by the vertical line passing through the x-coordinate of the vertex. Thus, the axis of symmetry will be \( x = 1 \).

Step 4: Determining the Domain and Range

The domain of a quadratic function is all real numbers since \( x \) can take any value. Thus the domain is \( (-\infty, \infty) \). The range of a quadratic function depends on the maximum or minimum value of the function. Since the coefficient of \( x^2 \) is positive in the given equation, the parabola opens upwards and the vertex is the minimum point of the parabola. Thus, the range of the function is \( [-4, \infty) \).

Step 5: Sketching the graph

The graph of the quadratic function is a parabola that opens upwards and has a vertex at the point (1, -4). It cuts the x-axis at (3, 0) and (-1, 0) and the y-axis at (0, -3). The graph is symmetric about the line \( x = 1 \), which is the axis of symmetry of the parabola.

Key Concepts

Vertex of a Parabola

When graphing a quadratic function, one of the key features to identify is the vertex of the parabola. The vertex represents the highest or lowest point on the graph, depending on the direction the parabola opens. For the quadratic function in the form of f(x) = Ax² + Bx + C, the x-coordinate of the vertex can be found using the formula h = -B/(2A).

Once h is calculated, the y-coordinate can be determined by evaluating the function at that point: k = f(h). Together, the coordinates (h, k) give us the vertex of the parabola. For instance, with the function f(x) = x² - 2x - 3, we find that the vertex is at the point (1, -4), which serves as the lowest point of the graph since the coefficient A is positive, indicating that the parabola opens upwards.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirroring halves. It passes through the vertex and has the equation x = h, where h is the x-coordinate of the vertex. In other words, it's the line that reflects the parabola onto itself. Knowing the axis of symmetry is integral when graphing quadratic functions as it helps to plot the parabola accurately and to understand its symmetrical properties.

For our example function f(x) = x² - 2x - 3, the axis of symmetry is the line x = 1. This line is a crucial aid for sketching the graph and for helping us predict the behavior and features of the parabola such as the location of the x-intercepts relative to the vertex.

Quadratic Function Domain and Range

The domain of a quadratic function refers to the set of all possible x-values that can be input into the function. For all quadratic functions, the domain is always all real numbers, represented by (-∞, ∞), since the parabola extends infinitely left and right. In contrast, the range refers to the set of possible y-values the function can output. It depends on whether the parabola opens up or down.

For upward opening parabolas like in our example f(x) = x² - 2x - 3, the vertex provides the minimum y-value. Therefore, the range starts from this minimum value and extends to positive infinity, which is represented as [k, ∞), where k is the y-coordinate of the vertex. Hence, the range for our function is [-4, ∞).

X-intercepts and Y-intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts, also known as roots or zeros, are points where the graph intersects the x-axis. To find them, we set the function equal to zero and solve for x. For the function f(x) = x² - 2x - 3, the x-intercepts are found at the points (3, 0) and (-1, 0). These are crucial for graphing as they indicate where the parabola touches or crosses the x-axis.

The y-intercept is found by evaluating the function at x = 0, revealing where the parabola intersects the y-axis. For the example function, the y-intercept occurs at (0, -3). Intercepts provide key points that can be used to sketch the overall shape of the parabola accurately on a graph.