Problem 63 - College Algebra with Corequisite Support

Question

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function. Contains (1,-3) and has the shape of \(f(x)=-x^{2}\). Vertex is on the \(y\) - axis.

Step-by-Step Solution

Answer

The equation is \(f(x) = -x^2 - 2\).

Step 1: Understand the Basic Form of the Quadratic Function

The given function is \(f(x) = -x^2\). This equation is in the standard form \(f(x) = ax^2\), where \(a = -1\). Since the problem states that the new function has the same shape as \(f(x)\), the value of \(a\) in the new function equation must also be \(-1\).

Step 2: Determine the Vertex Form of the Quadratic Equation

The vertex form of a quadratic equation is \(f(x) = a(x - h)^2 + k\), where \( (h, k) \) is the vertex of the parabola. Since the vertex is on the \(y\)-axis, \(h = 0\). Therefore, the vertex form for this equation is \(f(x) = -1(x - 0)^2 + k = -x^2 + k\).

Step 3: Substitute the Given Point into the Equation

We know the point \((1, -3)\) is on the curve. Substitute \(x = 1\) and \(f(x) = -3\) into the vertex form equation: \(-3 = -(1)^2 + k\).

Step 4: Solve for \(k\)

Simplify the equation from Step 3: \(-3 = -1 + k\). Solve for \(k\) by adding 1 to both sides: \(-3 + 1 = k\), which results in \(k = -2\).

Step 5: Write the Equation of the Quadratic Function

Now that we have \(k = -2\), substitute \(k\) back into the vertex form equation to get the final quadratic equation: \(f(x) = -x^2 - 2\).

Key Concepts

Vertex Form

The vertex form of a quadratic function is highly useful as it reveals important characteristics of its graph, namely, the vertex location and vertical stretch/compression. The vertex form is expressed as \( f(x) = a(x - h)^2 + k \). Here are some key points to understand:

• \( a \): This coefficient determines the opening direction and the width of the parabola. If \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• \( (h, k) \): This denotes the vertex of the parabola, effectively the turning point of the graph.
• The vertex form makes it easy to see transformations such as shifts or reflections in the graph.

To summarize, converting a quadratic equation to vertex form can simplify understanding its graph and making predictions about its behavior.

Standard Form

Quadratic functions often begin in standard form, which is represented as \( f(x) = ax^2 + bx + c \). This is the form with which many are familiar when learning about quadratics.

• \( a \): As in vertex form, this influences the parabola's direction and stretch/compression.
• \( b \): This influences the symmetry axis and point of intersection with other elements.
• \( c \): Represents the \( y \)-intercept, which is the point where the curve crosses the \( y \)-axis.

Standard form is advantageous when factoring the quadratic equation or finding the roots using the quadratic formula. It’s less intuitive than vertex form for visualizing graph transformations, but still essential for many algebraic operations.

Quadratic Equation

A quadratic equation is akin to a mathematical chameleon: it can appear in several forms but always represents the same parabolic curve. The general quadratic equation is in the form \( ax^2 + bx + c = 0 \). Understanding this equation involves:

• Recognizing that the highest degree (squared term) indicates a parabolic trajectory in the graph.
• Solving it can reveal the roots of the parabola, where it intersects the x-axis.
• Using various methods such as factoring, completing the square, or the quadratic formula to solve for the roots.

Quadratic equations are foundational in algebra, providing a gateway to understanding more complex polynomial equations and their corresponding graphs.

Parabola

A parabola is the U-shaped curve you get when graphing a quadratic function. It has many interesting properties and features:

• The axis of symmetry, a vertical line through the vertex, divides the parabola into two symmetrical halves.
• A parabola can open upward or downward, depending on the sign of the leading coefficient \( a \).
• The focus and directrix are concepts associated with the geometrical definition of a parabola, illustrating its unique reflective properties.

Understanding parabolas is crucial as they often represent real-world phenomena, such as the trajectory of projectiles.

Vertex

At the heart of a parabola lies its vertex, the apex of its curve. This point provides critical insight into the parabola's properties:

• The vertex is the highest or lowest point on the parabola, depending on whether it opens upwards or downwards.
• It serves as a turning point, changing the direction of the curve.
• In the vertex form \( f(x) = a(x - h)^2 + k \), \( (h, k) \) gives the coordinates of the vertex directly.

By understanding the vertex, you can immediately gather information about the behavior and position of the parabola on the coordinate plane.