Problem 97 - Algebra and Trigonometry

Question

graph each parabola with the given equation. $$ y=-3(x-1)^{2}+2 $$

Step-by-Step Solution

Answer

The parabola opens downwards with the vertex at (1, 2) and relative narrowness due to the coefficient of x^2 (-3).

Step 1: Identify the Vertex Form

The equation \(y = -3(x - 1)^{2} + 2\) is in the form \(y = a(x - h)^{2} + k\), which is a standard form for a parabola (also called the vertex form). In this form, (h, k) is the vertex of the parabola. Comparing the given equation to the standard form, we can say that a = -3, h = 1, and k = 2. So, the vertex of the parabola is (1, 2).

Step 2: Determine the Direction and Width

The coefficient of \(x^{2}\) (a = -3 here) determines whether the parabola opens upwards or downwards and also the width of the parabola. If 'a' is positive, the parabola opens upwards. If 'a' is negative (our case), the parabola opens downwards. Also, the value of |a| (absolute value) determines the width. Larger |a| means narrower parabola. Here, |a| = |-3| = 3, implying a relatively narrow parabola.

Step 3: Sketch the Parabola

Now it's time to graph our findings. Place the vertex at (1, 2) on the graph. Knowing the parabola is narrow and opens downwards helps in sketching the curve. Note that the parabola is symmetric with respect to the vertical line passing through the vertex (h, k).

Key Concepts

Quadratic Functions

Quadratic functions form the backbone of many algebraic applications, and they are commonly written in the form of \(y = ax^2 + bx + c\). This standard form expresses a polynomial equation of degree 2. However, when graphing, the vertex form \(y = a(x - h)^2 + k\) is often more useful.

This is because it clearly shows the location of the vertex

• \((h, k)\) is the vertex of the parabola and is crucial for graphing the quadratic function effectively.
• The coefficient \(a\) plays a significant role as it affects the shape and direction of the parabola.

When working with quadratic functions, transforming them to vertex form can make graphing simpler and more intuitive. It allows you to easily determine the maximum or minimum points of the graph, which are the vertex points.

Graphing Parabolas

Graphing a parabola involves plotting a smooth, symmetrical curve on the coordinate plane. To accurately graph a quadratic equation in vertex form like \(y = -3(x-1)^2 + 2\), you start with the vertex.

• Vertex Placement: Identify the vertex coordinates \((h, k)\), which are \((1, 2)\) for the given equation. Place this point on the graph.
• Determine the Shape: The sign and absolute value of \(a\) (\(-3\) here) dictate the parabola's shape. Since \(a\) is negative, the parabola opens downwards. The larger the absolute value, the narrower the curve.
• Symmetry: Parabolas are symmetrical around a vertical line that passes through the vertex. This can be used to sketch other points from the vertex.

Armed with these elements, you can draw the associated curve accurately, making sure it reflects the downward opening and narrow shape.

Vertex of a Parabola

The vertex is a fundamental component in understanding and graphing parabolas. For the equation \(y = -3(x - 1)^2 + 2\), the vertex has been identified as \((1, 2)\). The vertex forms the highest or lowest point of a parabola depending on whether it opens upwards or downwards.

The importance of the vertex includes:

• It is the turning point of the parabola, where the curve changes direction.
• In the equation, \((h, k)\) directly gives you the coordinates, making it unnecessary to complete the square or use the formula \(-b/(2a)\) for vertex calculation.
• The vertex helps in understanding the symmetrical properties of the parabola, which is most useful when sketching it.

By focusing on the vertex, you can efficiently graph and interpret quadratic functions, highlighting their geometric properties.