Problem 56 - 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 \((-2,-3) ;\) through \((0,-19)\)

Step-by-Step Solution

Answer

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

Step 1: Understanding 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, we know the vertex is \((-2, -3)\), so \(h = -2\) and \(k = -3\). Substitute these values into the vertex form to get \(P(x) = a(x + 2)^2 - 3\).

Step 2: Plug in the Known Point

The point \((0, -19)\) lies on the parabola. Substitute \(x = 0\) and \(P(x) = -19\) into the equation from Step 1: \(-19 = a(0 + 2)^2 - 3\).

Step 3: Solve for \(a\)

Continue with the equation \(-19 = a(2)^2 - 3\). Simplifying gives \(-19 = 4a - 3\). Add 3 to both sides to obtain \(-16 = 4a\). Divide both sides by 4 to find \(a = -4\).

Step 4: Write the Vertex Form with \(a\)

Substitute \(a = -4\), \(h = -2\), and \(k = -3\) back into the vertex form: \(P(x) = -4(x + 2)^2 - 3\).

Step 5: Expand to Standard Form

Now expand \(-4(x + 2)^2 -3 \) to express the quadratic in the standard form. Calculate \((x+2)^2 = x^2 + 4x + 4\), then multiply by \(-4\), giving \(-4x^2 - 16x - 16\). Finally, add \(-3\): \(P(x) = -4x^2 - 16x - 16 - 3\).

Step 6: Simplify to Standard Form

Combine like terms: \(P(x) = -4x^2 - 16x - 19\). Thus, the standard form of the quadratic function is \(P(x) = -4x^2 - 16x - 19\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is a powerful way to describe a parabola. This form is written as \( P(x) = a(x-h)^2 + k \), where \((h, k)\) represents the vertex of the parabola. Understanding the vertex form is crucial as it immediately gives us the vertex location which acts as the highest or lowest point of the parabola, depending on the value of \(a\).
- If \(a\) is positive, the parabola opens upwards. - If \(a\) is negative, the parabola opens downwards.
In this exercise, the given vertex is \((-2, -3)\), immediately informing us that \(h = -2\) and \(k = -3\). By substituting these into the vertex form, we have \(P(x) = a(x + 2)^2 - 3\). Knowing one additional point through which the parabola passes, like \((0, -19)\), allows us to solve for \(a\) and get a complete picture of the parabola's orientation and shape.

Standard Form

The standard form of a quadratic equation is expressed as \(P(x) = ax^2 + bx + c\). This form is useful for identifying the coefficients that affect the vertical stretch, the direction, and the shape of the parabola.
- \(a\) influences how "steep" or "flat" the parabola is. - \(b\) interacts with both \(a\) and \(c\) to determine the axis of symmetry. - \(c\) represents the y-intercept, where the parabola crosses the y-axis.
In the step-by-step solution, after determining \(a = -4\) from the vertex form, the problem expands this form to the standard form by multiplying out the squared term and simplifying any constants. This yields the equation \(P(x) = -4x^2 - 16x - 19\). Here, \(b = -16\) and \(c = -19\), allowing us to grasp how the parabola is situated in the coordinate plane.

Parabola Equation

The parabola equation connects various forms of quadratic equations, representing curves in an xy-plane. With any quadratic equation, you can express a parabola using multiple forms; most notably, the vertex form and the standard form.

• **Vertex Form**: Offers clarity about the parabola's vertex, helpful in graphing and understanding the extremity of the curve (the highest or lowest point).
• **Standard Form**: Convenient for algebraic manipulations, finding roots, and is instrumental for algorithms that compute intersection points, etc.

Ultimately, the utility of any particular form of a parabola equation depends on what specific information or operation is required. Transitioning from one form to another, as in the original exercise, allows students to enjoy the dual benefits of both clarity from vertex information as well as ease in solving and graph presentation from the standard form.