Problem 34 - 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. $$ f(x)=x^{2}+4 x-1 $$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=x^{2}+4x-1\) is a parabola that opens upwards with vertex at \((-2,-1)\), x-intercepts at \((-5, 0)\) and \((1, 0)\), and y-intercept at \((0,-1)\). Its axis of symmetry has the equation \(x=-2\). The function's domain is all real numbers, or \((-∞, ∞)\), and its range is all real numbers greater than or equal to -1, or \([-1, ∞)\).

Step 1: Find the Vertex

The vertex of a parabola of the form \(f(x) = ax^{2} + bx + c\) is given by \(-b/2a\) for the x-coordinate and \(f(-b/2a)\) for the y-coordinate. Here, \(a = 1\), \(b = 4\), \(c = -1\). The x-coordinate of the vertex is then \(-4/2(1) = -2\). Substituting \(-2\) into the equation gives us the y-coordinate: \((-2)^2+4(-2)-1 = -1\). So, the vertex is \((-2,-1)\).

Step 2: Find the Intercepts

The x-intercepts can be found by setting \(f(x)=0\) and solving for \(x\). The y-intercept can be found by setting \(x=0\) and solving for \(f(x)\). Doing so gives x-intercepts as \((-5,0)\) and \((1,0)\), and y-intercept as \((0,-1)\).

Step 3: Sketch the Graph

Plot the vertex and the intercepts on a graph. Remember, parabolas are symmetric, so the graph will be a U-shaped curve where the axis of symmetry goes through the vertex. Also, since the coefficient of \(x^2\) is positive, the parabola opens upwards.

Step 4: Determine the Axis of Symmetry

The axis of symmetry is a vertical line passing through the vertex. The x-coordinate of the vertex is always the equation of the axis of symmetry. Therefore, the axis of symmetry has the equation \(x=-2\).

Step 5: Determine the Domain and Range of the Function

The domain of a quadratic function is always all real numbers, or \((-∞, ∞)\). Since the parabola opens upward and has a lowest point (vertex) at \(y=-1\), the range of the function is all real numbers greater than or equal to -1, or \([-1, ∞)\).

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for students who are graphing quadratic functions. A parabola's vertex is the point where the curve turns; this can be either the highest point on the graph (if the parabola opens downward) or the lowest point (if it opens upward).

In the context of the function ( f(x)=x^2+4x-1 ), we determine the vertex by using the formula for the x-coordinate, (-b/2a), and then we find the y-coordinate by evaluating the function at that x value. Here, the vertex is (-2, -1), meaning the parabola turns at the point (-2,-1). This information serves as a starting point for sketching the graph of the function.

Axis of Symmetry

The axis of symmetry is a straight line that vertically splits the parabola into mirror images. Each point on one side of the axis at a given distance is reflected across to the other side at the same distance. For the parabola given by ( f(x)=x^2+4x-1 ), the axis of symmetry is the line x=-2. This means that the parabola is symmetric with respect to this line, and it is essential to plot this on the graph to make sure our parabola is accurate. Recognizing the axis of symmetry also helps in finding the x-intercepts of the function by understanding the balance of the curve.

Intercepts of a Quadratic Function

Intercepts are where the graph crosses the axes, providing valuable points for drawing the function's curve. For quadratic functions, there can be up to two x-intercepts and always one y-intercept.

For the function ( f(x)=x^2+4x-1 ), setting f(x) to zero lets us solve for the x-intercepts, resulting in points (-5,0) and (1,0). These represent where the parabola crosses the x-axis. The y-intercept is found by setting x to zero, giving us the point (0,-1), where the graph crosses the y-axis. Placing these intercepts on a graph aids in shaping the curve of the parabola.

Domain and Range of a Function

The domain and range of a function are concepts that describe what values the function can take as inputs (domain) and outputs (range).

For quadratic functions, like ( f(x)=x^2+4x-1 ), the domain is generally all real numbers, which we write as (-∞, ∞). This tells us that we can insert any real number into the function. The range, however, is determined by the direction the parabola opens and the position of its vertex. Since our parabola opens upwards and the vertex is at y=-1, the range is all real numbers greater than or equal to -1, written as [-1, ∞). This indicates the vertical extent of the graph—the values that f(x) can output.