Problem 27 - 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. \(f(x)=x^{2}-2 x-3\)

Step-by-Step Solution

Answer

The vertex is at (1, -4), the x-intercepts are at x = -1 and x = 3, the y-intercept is at y = -3, the axis of symmetry is \(x = 1\), the domain is all real numbers and the range is \(-4 \leq y < \infty\).

Step 1: Determine the Vertex

The vertex form of a parabola is \(y=a(x-h)^{2}+k\) where (h,k) are the coordinates of the vertex. Comparing this with the given equation \(f(x)=x^{2}-2x-3\), we need to convert the equation into vertex form. Complete the square on the equation. The form is \(f(x)=[x-(2/2)]^{2}-[(2/2)^2]-3 \Rightarrow f(x)=(x-1)^{2}-4 \). Now, we can see that the coordinates of the vertex are(1,-4).

Step 2: Determine the Intercepts

To determine the x-intercepts, we set \(f(x) = 0\), to get: \(0 = x^2 - 2x - 3 \). This can be factored to \(0 = (x - 3)(x + 1)\). Setting each factor equal to zero gives us the two x-intercepts: x = 3 and x = -1. The y-intercept can be found by substituting x = 0 into the equation, which gives: \(f(0) = (0)^2 - 2*(0) - 3 = -3\). Therefore, the y-intercept is -3.

Step 3: Determine the Axis of Symmetry

For a parabola of the form \(y = a(x - h)^2 + k\), the axis of symmetry is given by the equation x = h. Here, the equation of axis of symmetry is \(x = 1\).

Step 4: Determine the Domain and Range

The domain of the quadratic function is all real numbers because there is no restriction on the possible values of x. The range of a standard quadratic function is always the set of real numbers equal to or greater than the y-coordinate of the vertex. Here, with a vertex of (1, -4), the range of the function is \(-4 \leq y < \infty\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola plays a crucial role in graphing quadratic functions. It represents the highest or lowest point on the parabola, depending on its orientation. To find the vertex of a quadratic function, we can convert the equation into the vertex form, which is \( y = a(x-h)^2 + k \).
Here, \((h, k)\) are the coordinates of the vertex. Completing the square is often a reliable method to change the equation into this form.
In the example function \( f(x) = x^2 - 2x - 3 \), completing the square transforms it to \( f(x) = (x - 1)^2 - 4 \), pinpointing the vertex at \((1, -4)\).

• Vertex Form: Useful for determining the vertex directly.
• Example: For \( y = (x - 1)^2 - 4 \), the vertex is \((1, -4)\).

Intercepts

Intercepts are essential for understanding where the graph of a function crosses the axes. For quadratic functions, you can find the x-intercepts and the y-intercept:

• X-intercepts: To find them, set \( f(x) = 0 \) and solve for \( x \). The specific factored form of this polynomial, \((x - 3)(x + 1) = 0\), gives \( x \)-intercepts at \( x = 3 \) and \( x = -1 \).


• Y-intercept: This is where the graph crosses the y-axis, found by setting \( x = 0 \). For \( f(x) = x^2 - 2x - 3 \), substituting \( x = 0 \), results in \( f(0) = -3 \), so the y-intercept is \(-3\).

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It's centrally important for graphing and helps in identifying the symmetric nature of parabolic graphs.
For a quadratic function in vertex form, \( y = a(x-h)^2 + k \), the axis of symmetry is given by \( x = h \).
In our given example, the vertex form is \( f(x) = (x - 1)^2 - 4 \), making the axis of symmetry \( x = 1 \).

• Definition: A vertical line passing through the vertex.
• Equation: For the current parabola, it's \( x = 1 \).

Domain and Range

Understanding the domain and range of a quadratic function is vital to comprehending the extent and limitations of its graph.

• Domain: The domain of a quadratic function is all real numbers, primarily because there are no restrictions on the possible values that \( x \) can take. This means you can plug any real number into the function and get a valid output.


• Range: The range is all real numbers equal to or greater than the minimum value if the parabola opens upwards. For a function with a vertex \((h, k)\), where \( a > 0 \), the range is \( y \geq k \). In our example, since the vertex is \((1, -4)\), the range is \( y \geq -4 \).