Problem 69 - Algebra 2

Question

For each function, the vertex of the function's graph is given. Find \(a\) and \(b\) \ $$ y=a x^{2}+b x+8 ;(2,-4) $$

Step-by-Step Solution

Answer

The coefficients are \(a = 3\) and \(b = -12\).

Step 1: Identify the Vertex Form of a Quadratic

A quadratic function can be written in the vertex form as \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola. We can use the given vertex to write the vertex form as \(y = a(x - 2)^2 - 4\).

Step 2: Expand the Vertex Form

To find the values of \(a\) and \(b\) for the quadratic function \(y = ax^2 + bx + 8\), we expand \(y = a(x - 2)^2 - 4\) to match this standard form. Expand the square to get \(y = a(x^2 - 4x + 4) - 4\).

Step 3: Distribute the \(a\) and Adjust for the Vertex Constant

Distribute the \(a\) through the expanded square to obtain the standard form: \(y = ax^2 - 4ax + 4a - 4\).

Step 4: Compare Coefficients

Compare the expanded equation \(y = ax^2 - 4ax + 4a - 4\) with the given function \(y = ax^2 + bx + 8\). Equating coefficients, we can form the equations \(-4a = b\) and \(4a - 4 = 8\).

Step 5: Solve for \(a\)

Solve the equation \(4a - 4 = 8\) to find \(a\). Adding 4 to both sides gives \(4a = 12\), and dividing by 4 gives \(a = 3\).

Step 6: Solve for \(b\)

Use the equation \(-4a = b\) to find \(b\). Since we found that \(a = 3\), plug in to get \(-4\times3 = b\), which gives \(b = -12\).

Key Concepts

Vertex Form of a Quadratic

Understanding the vertex form of a quadratic function is critical for graphing and solving quadratic equations. The vertex form is expressed as \(y = a(x - h)^2 + k\), where \((h, k)\) represents the vertex of the parabola, and \(a\) determines the width and the direction of the parabola opening. If \(a > 0\), the parabola opens upwards, while \(a < 0\) indicates that it opens downwards.

A key aspect is to identify the vertex coordinates directly from the equation, which can be quite useful in graphing. In the given problem, we have the vertex coordinates \((2, -4)\), which tells us that the parabola crosses the vertex at this exact point on the coordinate plane.

Moreover, the vertex form makes it straightforward to find the axis of symmetry of the parabola, which is the vertical line \(x = h\), and to determine the maximum or minimum value of the quadratic function, which is \(k\) when \(x = h\). In our case, the axis of symmetry is \(x = 2\), and the minimum value of the function is \(-4\).

Expanding Quadratic Equations

To solve certain types of quadratic problems, especially those involving finding specific coefficients, it's useful to know how to expand a quadratic expression. Starting from the vertex form, we can expand the quadratic equation to transform it into its standard form \(y = ax^2 + bx + c\). This process involves squaring the binomial and then distributing the leading coefficient \(a\) across the terms within the parenthesis.

For instance, expanding \(y = a(x - 2)^2 - 4\) involves squaring \(x-2\) to get \(x^2 - 4x + 4\) and then multiplying each term by \(a\) to achieve the expanded form \(ax^2 - 4ax + 4a\). Finally, we incorporate the constant term \(-4\) to obtain \(ax^2 - 4ax + 4a - 4\).

This expanded form is pivotal as it directly relates to the coefficients in the standard form of the quadratic function. Being able to carry out this process accurately is essential when comparing coefficients, a method that forms the basis of finding unknowns in quadratic equations.

Coefficients Comparison

When working with quadratic equations, comparing coefficients is a technique that involves equating corresponding coefficients from two polynomial expressions that are equal to each other. In our exercise, after expanding the vertex form of the quadratic function and obtaining it in standard form, we compare this with the given quadratic function \(y = ax^2 + bx + 8\) to find the values of \(a\) and \(b\).

The equation from the expanded vertex form, \(y = ax^2 - 4ax + 4a - 4\), presents coefficients that must match those in the given function. By setting the coefficients of \(x^2\), \(x\), and the constant terms equal, we establish the system of equations from which we can solve for \(a\) and \(b\). Specifically, we equate \(-4a\) to \(b\) and \(4a - 4\) to \(8\).

Once you understand this method, you can easily solve for one variable and then use that value to find the other. This is a common and efficient approach to solving quadratic equations in algebra, and mastering it will support your mathematical problem-solving substantially.