Problem 52 - Precalculus : Building Concepts and Connections

Question

Find the vertex and axis of symmetry of the associated parabola for each quadratic function. Sketch the parabola. Find the intervals on which the function is increasing and decreasing, and find the range. $$h(x)=x^{2}+6 x-7$$

Step-by-Step Solution

Answer

The vertex of the parabola is at \((-3, -16)\), and the axis of symmetry is the line \(x=-3\). The function decreases on the interval \(-\infty,-3\) and increases on the interval \(-3,\infty\). The range of the function is \(-16,\infty)\).

Step 1: Finding the Vertex

For a given quadratic function in the form \(f(x) = a(x - h)^{2} + k\), \(h\) and \(k\) are the x and y-coordinates of the vertex respectively. They can be calculated using the formulas \(h = -\frac{b}{2a}\) and \(k = f(h)\). For \(h(x) = x^{2} + 6x - 7\), coefficients \(a = 1\) and \(b = 6\). Thus, the vertex coordinates are \(h = -\frac{6}{2 \cdot 1} = -3\) and \(k =(-3)^2+6(-3)-7 = -16\).

Step 2: The Axis of Symmetry and Sketching the Parabola

The axis of symmetry for a parabola is the vertical line \(x=h\), where \(h\) is the x-coordinate of the vertex. For this function, the axis of symmetry is \(x=-3\). To sketch the parabola, plot the vertex, draw the axis of symmetry, and then sketch the symmetric parabola.

Step 3: Intervals of Increase and Decrease

A parabola opens upwards if \(a>0\), and downward if \(a<0\). For this function, \(a=1\), which is greater than zero. Hence, the function declines as x approaches \(h\) from the left, and it rises as x moves away from \(h\) to the right. Therefore, the function decreases on the interval \(-\infty,-3\), and it increases on the interval \(-3, \infty\).

Step 4: Finding the Range

For an upwards opening parabola, the range is \(k,\infty)\), and for a downward-opening parabola, the range is \(-\infty,k\). For this function, it is \(-16,\infty)\).

Key Concepts

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that bisects the parabola into mirror images on each side. This axis passes through the vertex of the parabola, which is a significant feature since it represents the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards.

For the quadratic function, the axis of symmetry can be found using the formula \( x = -\frac{b}{2a} \), where \(a\) and \(b\) are the coefficients from the standard quadratic equation \( ax^2 + bx + c \). In the given function \(h(x) = x^2 + 6x - 7\), the axis of symmetry is at \( x = -3 \). This line divides the parabola such that for every point on one side of the axis, there is an identical point on the other side at the same distance from the axis.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, represented in the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and the value of \( a \) is non-zero. These functions graph as parabolas, which are U-shaped curves that can open upwards or downwards.

The shape and position of the parabola are determined by the coefficients \( a \), \( b \), and \( c \). The orientation (upwards or downwards) depends on the sign of \( a \), with positive \( a \) yielding an upward-opening parabola and negative \( a \) leading to a downward-opening parabola. Key features of quadratic functions include the vertex, axis of symmetry, intercepts, and the direction of the opening, which can all be deduced from the standard form of the equation or converted to vertex form.

Intervals of Increase and Decrease

In the context of quadratic functions, intervals of increase and decrease reveal the sections of the function where the output values (y-values) are getting larger (increasing) or smaller (decreasing) as the input values (x-values) change. These intervals are directly connected to the parabola's shape and the location of its vertex.

For an upward-opening parabola, the function will decrease until it reaches the vertex and then increase after the vertex. Conversely, for a downward-opening parabola, the function will increase until the vertex and then decrease after it. This behavior is due to the function's symmetry about the axis of symmetry that passes through the vertex. For the given function \(h(x) = x^2 + 6x - 7\), the function decreases on the interval \(-\infty, -3\) and increases on the interval \(-3, \infty\), where \(x = -3\) is the axis of symmetry.

Range of a Function

The range of a function is the set of all possible output values (y-values) that the function can produce. For quadratic functions, particularly parabolas, the direction in which the parabola opens significantly influences the range.

An upward-opening parabola, like the one represented by \(h(x) = x^2 + 6x - 7\), has a minimum point at the vertex. Hence, the range consists of all y-values greater than or equal to the y-coordinate of the vertex. The minimum value is the y-coordinate of the vertex, and every other possible y-value extends infinitely upwards. In this case, the range is \(-16, \infty\). Conversely, a downward-opening parabola would have a range where the y-values extend from negative infinity up to and including the y-coordinate of the vertex.