Problem 59 - 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 \((-4,-2) ;\) through \((2,-26)\)

Step-by-Step Solution

Answer

\(P(x) = -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3}\).

Step 1: Identify the Vertex Form

The vertex form of a quadratic function is given by \( P(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, the vertex is \((-4, -2)\), so \(h = -4\) and \(k = -2\). This gives us \( P(x) = a(x+4)^2 - 2 \).

Step 2: Substitute the Point

Substitute the coordinates of the point \((2, -26)\) into the equation \( P(x) = a(x+4)^2 - 2 \) to find \(a\).\[-26 = a(2+4)^2 - 2\].

Step 3: Solve for 'a'

Simplify and solve the equation for \(a\). \[-26 = a(6)^2 - 2\] \[-26 + 2 = 36a\] \[-24 = 36a\] \[a = \frac{-24}{36}\] \[a = -\frac{2}{3}\].

Step 4: Write the Vertex Form Equation

Substitute \(a = -\frac{2}{3}\) back into the vertex form equation: \[P(x) = -\frac{2}{3}(x+4)^2 - 2\].

Step 5: Expand to Standard Form

Expand \(P(x) = -\frac{2}{3}(x+4)^2 - 2\) to get the standard form. \[(x+4)^2 = x^2 + 8x + 16\] \[-\frac{2}{3}(x^2 + 8x + 16) - 2\] \[= -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{32}{3} - 2\] \[= -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3}\].

Step 6: Final Standard Form

The quadratic function in standard form is \[ P(x) = -\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3} \].

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to express quadratic equations, especially when you need to identify the vertex of the parabola quickly. It is described by the expression \( P(x) = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola. This form is useful when graphing a parabola because it explicitly shows the vertex, which is the highest or lowest point of the graph, depending on the value of \(a\). If \(a\) is positive, the parabola opens upwards; if \(a\) is negative, it opens downwards.

Understanding the vertex form provides insight into how the parabola is shifted horizontally and vertically in the coordinate plane. The vertex \((-4, -2)\) tells us that the parabola moves 4 units left and 2 units down from the origin. By identifying \(h\) and \(k\) as \(-4\) and \(-2\) respectively, the transformation from the parent function is clearly mapped. Students often find it easier to start with this form when analyzing or creating quadratic functions.

Standard Form

The standard form of a quadratic function is represented as \( P(x) = ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are coefficients. This form is particularly useful for calculations such as finding roots using the quadratic formula, the axis of symmetry, or performing further algebraic manipulations. It provides a straightforward way to determine the y-intercept of the parabola, which is the point where the graph intersects the y-axis, fixed at \(c\).

In the provided exercise, converting from vertex form to standard form involves binomial expansion and simplification. By expressing \((x+4)^2\) as \(x^2 + 8x + 16\), and distributing \(-\frac{2}{3}\), we arrive at the standard form \(-\frac{2}{3}x^2 - \frac{16}{3}x - \frac{38}{3}\). This transformation allows better analysis of the parabola's properties, such as finding where it crosses the x-axis by setting \(P(x) = 0\).

Understanding both forms helps students shift between different perspectives of the quadratic function, each highlighting distinct features.

Parabola

The graph of a quadratic function is known as a parabola, a symmetrical, curved shape that opens either upwards or downwards. The orientation and shape of the parabola are determined by the leading coefficient, \(a\). When \(a\) is negative, the parabola opens downwards, forming a concave shape like a frown. When \(a\) is positive, it opens upwards like a smile.

Key features of a parabola include the vertex, axis of symmetry, direction of opening, and the intercepts. The vertex is the peak or valley, the axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Understanding intercepts, where the parabola crosses the x-axis and y-axis, is crucial for sketching the graph.

Overall, mastering parabolas requires recognizing how transformations affect the shape and position of the graph in the coordinate plane. Comprehending these attributes enhances the ability to solve complex real-world problems modeled by quadratic functions.