Problem 65 - College Algebra

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 \((1,-6)\) has the shape of \(f(x)=3 x^{2}\) Vertex has \(x\) -coordinate of \(-1 .\)

Step-by-Step Solution

Answer

The quadratic function is \(g(x) = 3(x + 1)^2 - 18\).

Step 1: Analyze the given information

We need to find the quadratic function that passes through the point \((1, -6)\) and has the same shape as the function \(f(x) = 3x^2\). The shape indicates that the function has the same leading coefficient, which is 3, but its vertex is shifted. The vertex of this parabola lies at \(x = -1\).

Step 2: Write the general form of a parabola with given vertex

Since the vertex form of a quadratic function is given by \(g(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex, we substitute \(h = -1\) to get \(g(x) = 3(x + 1)^2 + k\). We use this form because the function maintains the shape of \(f(x) = 3x^2\).

Step 3: Substitute the point into the equation to find \(k\)

We know \((1, -6)\) is a point on the parabola. Substitute \(x = 1\) and \(g(x) = -6\) in the equation \(g(x) = 3(x + 1)^2 + k\):\[-6 = 3(1 + 1)^2 + k\]

Step 4: Solve for \(k\)

Compute inside the parentheses, square it, multiply by 3, and solve for \(k\):\[-6 = 3(2)^2 + k\] \[-6 = 3(4) + k\] \[-6 = 12 + k\] \[k = -6 - 12\] \[k = -18\]

Step 5: Write the final equation

Substitute \(k = -18\) into the vertex form equation to get the equation of the quadratic function: \[g(x) = 3(x + 1)^2 - 18\]. This is the equation of the quadratic function that includes the point \((1, -6)\) and follows the shape of \(f(x) = 3x^2\) with a vertex \(x\)-coordinate of \(-1\).

Key Concepts

Vertex Form

In the world of quadratic functions, the vertex form is a specific way to express the equation of a parabola. It is extremely useful when you are given the vertex of the parabola or need to identify it quickly. This form of the equation is written as \(g(x) = a(x - h)^2 + k\), where:

• \(a\) determines the width and the direction of the parabola's opening.
• \((h, k)\) is the vertex of the parabola, which is the highest or lowest point of the curve.

Knowing the vertex provides insight into the parabola's "turning point," which is crucial for graphing. In our specific exercise, we worked with a vertex at \((h, k) = (-1, -18)\). This means our parabola was shifted left by 1 unit and down by 18 units. Hence, the vertex is a central feature that helps in sketching or understanding the behavior of the graph.

Parabola

A parabola is a U-shaped curve that you often encounter in mathematics, physics, and engineering. It is the graph of a quadratic function, which is any function that can be expressed in the form \(ax^2 + bx + c\). In this problem, we wrote our equation in vertex form, which highlights two main points:

• The vertex, which is the minimum or maximum point of the parabola.
• Its symmetry, meaning that the parabola looks the same on both sides of the vertex.

Our parabolas are typically symmetrical along a line called the "axis of symmetry," which passes through the vertex. For our problem, because the vertex was at \((-1, -18)\), the axis of symmetry was the vertical line \(x = -1\). Understanding these characteristics helps in predicting and explaining the motion or path of objects that follow a parabolic trajectory.

Leading Coefficient

In a quadratic equation, the leading coefficient is the number "a" in the term \(ax^2\), which affects the parabola's direction and width. This number plays a significant role:

• If \(a > 0\), the parabola opens upwards, making a U-shape.
• If \(a < 0\), it opens downwards, like an upside-down U.
• The value of \(a\) also affects the width; larger \(|a|\) values make the parabola narrower, while smaller \(|a|\) values make it wider.

In the exercise, the given function \(f(x) = 3x^2\) has a leading coefficient of 3. This same coefficient guided the shape of the new function we derived, \(g(x) = 3(x + 1)^2 - 18\). The positive coefficient tells us the parabola opens upwards, keeping a similar shape to the original function but shifted in position. Understanding the leading coefficient is vital, as it helps anticipate how changes to the equation alter the graph.