Problem 62 - Algebra 1

Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-\frac{1}{3} x^{2}+2 x-3 $$

Step-by-Step Solution

Answer

The graph is a parabola that opens downwards, with its vertex at the point (3, -2).

Step 1: Convert the given function into vertex form

To convert into the vertex form, use the formula \( h = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Then substitute \( h \) into the standard function to find \( k \), the y-coordinate of the vertex. Here \( a = -\frac{1}{3} \), \( b = 2 \) and \( c = -3 \). We get \( h = -\frac{2}{2*(-1/3)} = 3 \). Substituting \( x = h \) in the function, we get \( y = -\frac{1}{3}*3^{2}+2*3-3 = -2 \). Therefore, the vertex is at (3, -2).

Step 2: Identify the direction of the parabola

The direction of the parabola depends on the coefficient of \( x^{2} \) in the quadratic function. If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards. Here \( a = -\frac{1}{3} \), so the parabola opens downwards

Step 3: Plot the vertex and draw the parabola

Plot the vertex at the coordinate (3,-2) on graph paper. Since the parabola opens downwards, draw a parabola opening downwards from the vertex. The sketch should be a rough estimate of the curve's shape. The more points plotted, the more accurate the graph.

Key Concepts

Vertex Form

Understanding the vertex form of a quadratic function is essential in sketching its graph. A quadratic function can be expressed in the vertex form as \( y = a(x-h)^2 + k \). In this format, \((h, k)\) represents the vertex of the parabola. The parameter \(a\) influences the width and direction of the parabola. By rewriting a quadratic equation in vertex form, it becomes easier to identify the vertex and helps in sketching the graph quickly. To convert a standard quadratic function \(y=ax^2+bx+c\) into vertex form, calculate \(h\) using the formula \( h = -\frac{b}{2a} \). Then, find \(k\) by substituting \(h\) back into the function to solve for \(y\). This method provides the exact location of the vertex, making it a simple task to plot it on the graph.

Quadratic Function

A quadratic function is a type of polynomial function that has the form \( y = ax^2 + bx + c \). It's characterized by its graph, which is a parabola. Key components of a quadratic function include:

• The coefficient \(a\), which dictates how "wide" or "narrow" the parabola appears.
• The term \(b\), contributing to the direction and position of the parabola.
• The constant \(c\), indicating where the parabola intersects the y-axis.

Studying quadratic functions is pivotal in math because they model various real-world phenomena. Understanding their graphs not only helps in solving equations but also in interpreting data where the relationship between the variables forms a parabolic shape.

Parabola Direction

The direction in which a parabola opens is determined by the sign of the coefficient \(a\) in the quadratic function \(y = ax^2 + bx + c\). If \(a > 0\), the parabola opens upwards. On the other hand, if \(a < 0\), the parabola opens downwards. This direction is crucial when sketching the parabola, as it affects the overall shape and the position of its vertex relative to the graph. In practical terms, if the parabola opens upwards, it's like a "smile" and will have a minimum point at the vertex. Conversely, if it opens downwards, it resembles a "frown" with a maximum point at the vertex. Understanding these nuances can aid in accurately sketching these graphs on paper.

Graph Sketching

Graph sketching involves translating the components of a quadratic function onto a visual format. Here's a simple approach to sketching:

• Start by plotting the vertex, which serves as the central point of the parabola.
• Identify whether the parabola opens upwards or downwards by examining the coefficient \(a\).
• Make use of symmetry, as parabolas are symmetric along the vertical line passing through the vertex.
• Plot additional points on either side of the vertex to determine the width and shape.

Remember, the sketch doesn't have to be perfect but should accurately reflect the features of the function, including its vertex, direction, and intersection points. With practice, graph sketching becomes an intuitive and valuable skill in visualizing quadratic equations.