Problem 27 - College Algebra with Corequisite Support

Question

For the following exercises, use the vertex \((h, k)\) and a point on the graph \((x, y)\) to find the general form of the equation of the quadratic function. $$ (h, k)=(-2,-1),(x, y)=(-4,3) $$

Step-by-Step Solution

Answer

The general form of the quadratic equation is \( f(x) = x^2 + 4x + 3 \).

Step 1: Understand the Vertex Form of a Quadratic Equation

The vertex form of a quadratic equation is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, \(h = -2\) and \(k = -1\). We need to express the equation in this form initially.

Step 2: Substitute the Vertex into the Vertex Form

Plug the vertex \((-2, -1)\) into the vertex form equation, giving us: \( f(x) = a(x + 2)^2 - 1 \). Notice that \((x - h)\) becomes \((x + 2)\) due to \(h = -2\).

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

We know that the point \((-4, 3)\) lies on the graph of the function, meaning when \(x = -4\), \(f(x) = 3\). Substitute \(x = -4\) and \(f(x) = 3\) into the equation: \( 3 = a(-4 + 2)^2 - 1 \). Solve for \(a\): \( 3 = a(2)^2 - 1 \) \( 3 = 4a - 1 \) \( 4 = 4a \) \( a = 1 \).

Step 4: Write the Vertex Form with 'a'

With \(a = 1\), substitute back into the vertex form equation: \( f(x) = 1(x + 2)^2 - 1 \). Simplifying gives us: \( f(x) = (x + 2)^2 - 1 \).

Step 5: Convert to General Form

Expand \((x + 2)^2 - 1\) to convert the equation to general form: \( f(x) = (x^2 + 4x + 4) - 1 \) Leading to the general form: \( f(x) = x^2 + 4x + 3 \).

Key Concepts

Vertex Form

The vertex form of a quadratic equation is a popular way of expressing quadratics because it highlights the vertex, a crucial geometric feature of the parabola. This form is written as \( f(x) = a(x - h)^2 + k \). Here, \((h, k)\) represents the vertex, and \(a\) is a constant that affects the width and direction of the parabola.
When using the vertex form, you get immediate insight into the parabola's vertex, which is the point \((h, k)\) where the parabola either reaches a maximum or a minimum value, depending on whether \(a\) is negative or positive.
To write a quadratic equation in vertex form, you must know both the vertex and another point on the parabola. This additional point helps determine the value of \(a\), which scales the parabola and affects its orientation.

General Form of Quadratic Equation

The general form of a quadratic equation is expressed as \( ax^2 + bx + c \), where \(a\), \(b\), and \(c\) are constants. This form is complete and straightforward, allowing for easy calculation of the values for specific \(x\) inputs.
It's notably used in mathematical applications, such as factoring, solving quadratic equations using the quadratic formula, and finding the x-intercepts or roots of the function.

• \(a\) determines the direction and the "width" of the parabola.
• \(b\) controls the location of the axis of symmetry.
• \(c\) represents the y-intercept of the quadratic function on a graph.

After determining \(a\) using a point and the vertex form, expanding and simplifying the equation allows for conversion to the general form, making it useful for broader analysis of the quadratic function's behavior.

Parabola Vertex

The vertex of a parabola is a pivotal point on its curve, serving as either the lowest or highest point depending on the parabola's orientation.
In the vertex form \( f(x) = a(x - h)^2 + k \), the coordinates of the vertex are readily available as \((h, k)\). The parameter \(h\) indicates a left or right shift from the y-axis, while \(k\) denotes an upward or downward shift from the x-axis.
Understanding the vertex’s role is essential when sketching a parabola or determining its primary features such as maximum, minimum, and axis of symmetry.
In our exercise, the vertex \((-2,-1)\) reveals that the parabola shifts 2 units left and 1 unit down from the origin. This immediate insight simplifies forecasting the parabola’s behavior and modifications needed when altering \(a\) or shifting to the general form.