Problem 68 - College Algebra

Question

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose y-coordinate is the same as the given point. \(f(x)=3(x+2)^{2}-5 ; \quad(-1,-2)\)

Step-by-Step Solution

Answer

The axis of symmetry for the given parabola is \(x = -2\). And the second point on the parabola, having the same y-coordinate as (-1,-2), is (-3,-2).

Step 1: Determine the Axis of Symmetry

The formula for the axis of symmetry in the standard form of a parabola (which is \(f(x) = a(x-h)^2 + k\)) is \(x = h\). In the provided equation, we have \(f(x) = 3(x+2)^2 - 5\), from which we can see that h = -2. So, the axis of symmetry can be defined as \(x = -2.\)

Step 2: Comparing the Given Point to the Axis of Symmetry

We have our point (-1, -2) and we now know the axis of symmetry is \(x = -2\). We can notice that the x-coordinate of our point is 1 unit to the right of the axis of symmetry. So, to find a second point with the same y-coordinate, we can go 1 unit to the left of the axis of symmetry, giving us a new point at (-3, -2).

Step 3: Verification

We can substitute x = -3 into the parabola equation to verify whether or not its corresponding y-coordinate is -2. For the expression \(f(-3) = 3(-3 + 2)^2 - 5\), the result is indeed -2, confirming our solution is correct.

Key Concepts

Parabola

A parabola is a symmetric curve that has the shape of a U or an inverted U, commonly studied in algebra and geometry. It can represent various real-life phenomena like the path of a projectile under the influence of gravity.

Every parabola is characterized by a few key components: the vertex, the focus, the directrix, and the axis of symmetry. The vertex is the highest or lowest point on the curve, depending on whether the parabola opens upwards or downwards. The focus is a point from which distances to any point on the parabola and its corresponding point on the directrix are equal. When we have an equation of a parabola given by a function like in our exercise, the parabola approach allows us to quickly analyze these components and the shape of the graph.

Standard Form of a Parabola

The standard form of a parabola's equation is expressed as \(f(x) = a(x-h)^2 + k\), where \( (h, k) \) is the vertex of the parabola, and \(a\) determines the direction of opening (up or down) and the width of the parabola. If \(a\) is positive, the parabola opens upwards, and if negative, it opens downwards. The larger the absolute value of \(a\), the narrower the parabola.

The equation \(f(x) = 3(x+2)^2 - 5\) from our example is in the standard form where \(h = -2\) and \(k = -5\). Thus, the vertex of the parabola is at the point \((-2, -5)\), and since \(a = 3\) is positive, our parabola opens upwards with a relatively narrow spread.

Symmetry in Algebra

Symmetry in algebra is a property that indicates balance and equality about a central line or point. The axis of symmetry of a parabola is a vertical line that passes through its vertex and divides it into two mirror images. For the parabola described by \(f(x) = a(x-h)^2 + k\), the axis of symmetry is \(x = h\).

In our exercise, the axis of symmetry for the parabola \(f(x) = 3(x+2)^2 - 5\) is \(x = -2\), which is deduced from comparing the function to its standard form. This axis allows us to find the mirror image of a point across it, as exemplified in the step-by-step solution by finding the symmetrical point of \((-1, -2)\), which ended up being \((-3, -2)\).

Quadratic Functions

Quadratic functions represent equations of the second degree, typically written in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero. These functions graph as parabolas on the Cartesian plane. The solutions to the quadratic equation \(ax^2 + bx + c = 0\) are known as the roots or zeros of the function and they correspond to the x-intercepts of the parabola.

The focus of a quadratic function is not just to find its roots but also to analyze its graph's shape, the direction it opens, and its vertex and axis of symmetry. Transforming the quadratic function into its standard form is a helpful step in revealing these characteristics, just as our exercise illustrates by finding the axis of symmetry to help locate another point on the parabola.