Problem 32 - 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. $$ (h, k)=(0,1),(x, y)=(1,0) $$

Step-by-Step Solution

Answer

The general form of the quadratic equation is \( f(x) = -x^2 + 1 \).

Step 1: Recognize the Vertex Form of a Quadratic Equation

A quadratic function can be expressed in vertex form as \( f(x) = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, \(h = 0\) and \(k = 1\). Thus, the equation becomes \( f(x) = a(x-0)^2 + 1 = a(x)^2 + 1 \).

Step 2: Use the Given Point to Find the Value of 'a'

Substitute the point \((x, y) = (1, 0)\) into the equation \( f(x) = a(x)^2 + 1 \). So, \( 0 = a(1)^2 + 1 \) which simplifies to \( 0 = a + 1 \). Solve this equation for \(a\) by subtracting 1 from both sides: \( a = -1 \).

Step 3: Write the Finite Quadratic Equation in Vertex Form

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

Step 4: Convert to General Form

Expand the equation \( f(x) = -1(x)^2 + 1 \) to get it into general form. This simplifies directly to \( f(x) = -x^2 + 1 \). The general form of a quadratic equation is \( ax^2 + bx + c \). Here, \( a = -1, b = 0, c = 1 \) so the equation is already in general form: \( -x^2 + 1 \).

Key Concepts

vertex form

A quadratic equation can be conveniently expressed in vertex form. The vertex form is given by the formula: \( f(x) = a(x - h)^2 + k \). Here:

• \(a\) is a coefficient that affects the direction and the width of the parabola.
• \((h, k)\) is the vertex of the parabola.

The vertex form is helpful because it easily shows the vertex of the parabola, which is either its highest or lowest point, depending on the value of \(a\). If \(a\) is positive, the parabola opens upwards (U-shape). If \(a\) is negative, the parabola opens downwards (inverted U-shape). In our exercise, the vertex is at \((0, 1)\), so the vertex form begins as \(f(x) = a(x - 0)^2 + 1\). This simplifies to \(f(x) = a(x)^2 + 1\). The vertex form makes it easy to plot the parabola and understand its transformations.

general form

The general form of a quadratic function is \( ax^2 + bx + c \). It's the standard way of writing quadratic equations and has these parts:

• \(a\) is the coefficient of \(x^2\), which indicates the parabola's direction (up if \(a > 0\), down if \(a < 0\)).
• \(b\) determines the axis of symmetry of the parabola.
• \(c\) is the function's y-intercept.

Converting from other forms, such as vertex form, to this standard form involves expanding and simplifying the equation. For the exercise at hand, starting from the vertex form \(f(x) = -1(x)^2 + 1\), we can directly see it's already simplified into the general form: \(-x^2 + 1\). Here we find \(a = -1\), \(b = 0\), and \(c = 1\). The general form is widely used in algebra and calculus, and it's the form used for most algebraic manipulations like factorization.

quadratic function

A quadratic function describes a parabola and is usually given by the formula \(f(x) = ax^2 + bx + c\). It's called 'quadratic' because it involves squaring, as \(x^2\) is the highest power of \(x\) in the equation. Quadratic functions are essential in mathematics and appear in various contexts such as physics, engineering, and economics.
The key components of a quadratic function are:

• The "leading coefficient" \(a\), which determines the parabola's openness and direction.
• The "linear coefficient" \(b\), which controls the tilt and axis of symmetry.
• The "constant term" \(c\), which gives the point where the graph intersects the y-axis.

These functions graph into parabolas which can model real-world scenarios like projectile paths. Understanding how to transform and work with these functions is crucial for solving various mathematical problems.

parabola

A parabola is the graphical representation of a quadratic function. Its defining feature is its U-shaped curve, which can open either upwards or downwards:

• An upward-opening parabola occurs when \(a > 0\).
• A downward-opening parabola occurs when \(a < 0\).

Parabolas are not limited to one specific orientation but can also "face" left or right, depending on how the quadratic equation is written and manipulated.
The vertex of a parabola is a crucial point, which either represents the minimum or maximum value of the quadratic function, based on its direction. The vertex gives you a clear reference point for shifts and transformations of the graph. In our problem, the vertex is \((0, 1)\), marking the highest point of the downward-opening parabola given \(a = -1\).
Parabolas have practical applications, such as in satellite dishes and headlights, where their reflective properties are harnessed. They also appear in various physics problems, where they describe paths and acceleration patterns.