Problem 57 - A Graphical Approach to Precalculus with Limits

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 \((8,3) ;\) through \((10,5)\)

Step-by-Step Solution

Answer

The equation is \(P(x) = \frac{1}{2}x^2 - 8x + 35.\)

Step 1: Identify the Vertex Form

We are given that the vertex of the quadratic function is \((8,3)\). In the vertex form of a quadratic equation, \(P(x) = a(x-h)^2 + k\), the vertex \((h,k)\) directly gives us \(h=8\) and \(k=3\). So, our equation becomes \(P(x) = a(x-8)^2 + 3\).

Step 2: Substitute the Point into the Vertex Form

The quadratic function passes through the point \((10, 5)\). Substituting \(x = 10\) and \(P(x) = 5\) into the equation \(P(x) = a(x-8)^2 + 3\), we have:\[5 = a(10-8)^2 + 3.\]

Step 3: Solve for the Parameter 'a'

Simplifying the equation from Step 2 gives:\[5 = a(2)^2 + 3 \]\[5 = 4a + 3.\]Subtract 3 from both sides to isolate the term with 'a':\[2 = 4a.\]Divide by 4 to find \(a\):\[a = \frac{1}{2}.\]

Step 4: Write the Quadratic in Vertex Form

Now that we know \(a=\frac{1}{2},\) the vertex form of the equation is:\[P(x) = \frac{1}{2}(x-8)^2 + 3.\]

Step 5: Convert to Standard Form

To express this in the standard form \(P(x) = ax^2 + bx + c\), expand \(P(x) = \frac{1}{2}(x-8)^2 + 3\).First, expand \((x-8)^2\):\[(x-8)^2 = x^2 - 16x + 64.\]Next, distribute \(\frac{1}{2}\):\[P(x) = \frac{1}{2}(x^2 - 16x + 64) + 3.\]\[P(x) = \frac{1}{2}x^2 - 8x + 32 + 3.\]Combine like terms:\[P(x) = \frac{1}{2}x^2 - 8x + 35.\]

Key Concepts

Vertex Form

The vertex form of a quadratic function provides an insightful way to understand the structure of quadratic equations. This form is denoted as \( P(x) = a(x-h)^2 + k \). Here, the vertex of the parabola \((h, k)\) gives the point where the parabola either peaks or dips, depending on whether it opens upwards or downwards.

• The value \(a\) determines the direction and the steepness of the parabola. If \(a\) is positive, the parabola opens upwards; if negative, it opens downwards.
• The vertex, \((h, k)\), serves as a turning point of the parabola. In our example, this was at \((8, 3)\).

To find the equation using the vertex form, you insert the known vertex and solve for other variables when an additional point through which the parabola passes is provided. This approach is particularly handy when graphing, as the vertex gives you a clear starting point to sketch the curve.

Standard Form

The standard form of a quadratic function is expressed as \( P(x) = ax^2 + bx + c \). This format is beneficial for understanding and predicting the parabola's overall behavior on a graph.

• The coefficients \(a\), \(b\), and \(c\) determine the parabola's opening, width, and vertical position, respectively.
• The leading coefficient \(a\) determines if the parabola opens upwards (when positive) or downwards (when negative).

To convert from the vertex form to this standard form, you need to expand the squared term \((x-h)^2) \) then simplify the expression through distribution and combining like terms.This conversion gives a more algebraic representation, allowing for straightforward calculations and solving quadratic equations through factorization or using the quadratic formula.

Vertex

In quadratic functions, the vertex \((h, k)\) is a crucial element that provides important information about the graph's shape and orientation. This point represents the maximum or minimum value of the function, depending on whether the parabola opens upwards or downwards.

• In the vertex form equation \( P(x) = a(x-h)^2 + k \), the vertex is directly extracted as \((h, k)\).
• The horizontal location \(h\) indicates where along the x-axis the vertex lies.
• The vertical coordinate \(k\) marks how far up or down this point is on the y-axis.

Knowing the vertex is essential for graphing as it helps to determine the symmetry axis of the parabola at \(x = h\). This single point helps elucidate the overall curve of the quadratic graph, guiding you through plotting additional points evenly around it.