Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=-3 x^{2}+6 x-9 $$

Short answer

The graph of \(y = -3x^2 + 6x - 9\) is a downward opening parabola with the vertex at the point (1, -6).

Step-by-step solution

Identify the vertex

Calculate the x-coordinate of the vertex using the formula \(-b/2a\), where \(b = 6\) and \(a = -3\), we get \(x = -6/(-2*3) = 1\). The y-coordinate is found by substituting x into the given equation, \(y = -3*(1)^2 + 6*1 - 9 = -6\). Hence, the vertex of the graph is (1, -6).

Sketch the Graph

The graph of the given quadratic function opens downwards as the coefficient of \(x^2\) is negative (-3). Start by plotting the vertex at (1, -6). Since this is a standard parabola scaled by the factor -3, points to the left and right of the vertex move downwards. The graph would appear to be a parabola inverted, with its vertex at (1, -6). To get other points on the graph, you can create a table of values for \(y = -3 x^2 + 6 x - 9\), by substituting random x-values.

Label the Vertex

The last step after drawing the curve is to label the vertex point. Indicate the point (1, -6) on the graph and call it 'Vertex'.This gives anyone looking at the graph a reference point to understand how it is shaped.

Key concepts

Vertex of a Parabola

The vertex of a parabola is a crucial point, as it represents the highest or lowest point on the graph. It's where the parabola changes direction. For the quadratic function in the form of y = ax^2 + bx + c, the vertex can be found using the formula x = -b/(2a). In our example, with the given function y = -3x^2 + 6x - 9, setting a = -3 and b = 6, the x-coordinate of the vertex is calculated as x = -6/(-2*3) = 1.

After determining the x-coordinate, we plug this value back into the equation to find the y-coordinate. For our vertex, substituting x = 1 gives us y = -3(1)^2 + 6*1 - 9 = -6, thus the vertex coordinates are (1, -6). Knowing the vertex is helpful as it serves as a starting point for sketching the graph and understanding the parabola's orientation and width.

Quadratic Functions

Quadratic functions are a specific type of polynomial function with the highest power of the variable being a square; they can be written in the general form y = ax^2 + bx + c. The coefficient a determines the direction of the parabola: if a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards.

In the context of our exercise, the coefficient a = -3 indicates that the parabola will open downwards. The quadratic function represents a variety of physical phenomena, such as projectile motion and the shape of satellite dishes. Its graph provides a visual representation of the possible values it can take. One can construct a table of values by selecting x-points and calculating the corresponding y-values to plot the parabola's shape effectively.

Graphing Parabolas

Graphing parabolas is a systematic process that begins with identifying key features such as the vertex, axis of symmetry, and direction of opening. For the function y = -3x^2 + 6x - 9, we start by plotting the vertex, determined as (1, -6). The axis of symmetry is a vertical line that passes through the vertex, specifically at x = 1 in this case.

To gather more points for the graph, we can create a table of values by substituting x-values into the equation. However, we can infer more points by symmetry about the axis. Since the leading coefficient is -3, each plotted point from the vertex stretches further downwards compared to a parabola with a = -1, suggesting it is narrower. Once points are plotted, we can draw a smooth curve through them, ensuring the parabola opens downwards due to the negative a. Lastly, the vertex's labeling provides a clear reference for the graph's shape and direction, marking the point (1, -6) as 'Vertex'.