Problem 56 - A Graphical Approach to College Algebra

Question

Find the equation of the quadratic function satisfying the given conditions. (Hint: Determine values of \(a\), \(h,\) and \(k\) that satisfy \(P(x)=a(x-h)^{2}+k .\) ) Express your answer in the form \(P(x)=a x^{2}+b x+c .\) Use your calculator to support your results. Vertex \((-2,-3) ;\) through \((0,-19)\)

Step-by-Step Solution

Answer

The equation of the quadratic function is \( P(x) = -4x^2 - 16x - 19 \).

Step 1: Understand the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by the formula \( P(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. In this exercise, the vertex given is \((-2, -3)\), so \( h = -2 \) and \( k = -3 \).

Step 2: Plug Vertex Values into Vertex Form

Substitute \( h = -2 \) and \( k = -3 \) into the vertex form equation: \[ P(x) = a(x + 2)^2 - 3 \].

Step 3: Use the Given Point to Find 'a'

We also know the parabola passes through the point \((0, -19)\). Substitute \( x = 0 \) and \( P(x) = -19 \) into the equation to find \( a \): \[ -19 = a(0 + 2)^2 - 3 \] Calculate: \[ -19 = a(4) - 3 \]\[ -19 = 4a - 3 \]\[ -16 = 4a \]\[ a = -4 \].

Step 4: Rewrite the Vertex Form with Found 'a'

Replace \( a \) with \(-4\) in the vertex form equation: \[ P(x) = -4(x + 2)^2 - 3 \].

Step 5: Expand the Vertex Form to Standard Form

Expand \( (x + 2)^2 \) to convert the equation to its standard quadratic form:\[ P(x) = -4(x^2 + 4x + 4) - 3 \].Distribute \(-4\):\[ P(x) = -4x^2 - 16x - 16 - 3 \].Combine like terms:\[ P(x) = -4x^2 - 16x - 19 \].

Key Concepts

Vertex Form

The vertex form of a quadratic function is an elegant way to understand key characteristics of the parabola quickly. It is given by the equation \( P(x) = a(x-h)^2 + k \), where \( (h, k) \) is the vertex. The vertex of a parabola represents its highest or lowest point, depending on the direction the parabola opens. This point is also where the parabola changes direction. In the context of our exercise, we are given a vertex at \((-2, -3)\). Here, \(h = -2\) and \(k = -3\), which means our parabola shifts 2 units to the left and 3 units down from the origin in the coordinate plane. This makes it easy to visualize the parabola's peak or trough.

• When \(a > 0\), the parabola opens upwards, and the vertex is the lowest point.
• When \(a < 0\), as in our exercise, the parabola opens downwards, and the vertex is the highest point.

Knowing the vertex immediately tells us a lot about the parabola's position and shape, making the vertex form particularly useful in both graphing and analyzing a parabola's behavior.

Standard Form

The standard form of a quadratic function is another common expression format, defined by \( P(x) = ax^2 + bx + c \). It reveals different insights like the parabola's y-intercept and a quick way to start derivative calculation in calculus. In standard form:

• \(a\) determines the direction and the width of the parabolic curve. A larger absolute value indicates a narrow parabola, while a smaller value is a wider one.
• \(b\) affects the direction and placement on the x-axis, impacting the parabola's horizontal placement.
• \(c\) represents the y-intercept, the point where the parabola crosses the y-axis.

For our exercise, after expanding the vertex form, we reached the equation \( P(x) = -4x^2 - 16x - 19 \). Here, \(c = -19\) means the parabola crosses the y-axis at \(-19\). The process of converting from vertex to standard form involves simple expansion and combining like terms.

Parabola

A parabola is a U-shaped curve that can open upwards or downwards on the coordinate plane, depending on the sign of \(a\) in the quadratic equation. The vertex forms the pinnacle or nadir of this shape. Parabolas have several interesting properties:

• The axis of symmetry is a vertical line that passes through the vertex, given by \(x = h\). It divides the parabola into two mirror-image halves.
• The focus is a point inside the parabola that helps define its shape, and every point on the parabola is equidistant to a line called the directrix and this focus.
• As mentioned before, when \(a > 0\), the parabola opens upwards, while if \(a < 0\), it opens downwards.

Understanding these characteristics helps in graph drawing and analyzing motions like projectile paths, optics, and more. In our exercise, the parabola opens downwards due to \(a = -4\), and its highest point lies at the vertex \((-2, -3)\). This knowledge is crucial when interpreting the function's behavior over its domain.