Problem 51 - Algebra 1

Question

SKETCHING GRAPHS Sketch the graph of the function. Label the vertex. $$ y=6 x^{2}-4 x $$

Step-by-Step Solution

Answer

The graph of the function \(y = 6x^{2} - 4x\) is an upward-opening parabola with vertex at \((1/3, -2/3)\).

Step 1: Identify the form of the quadratic function

The general form of any quadratic function is \(y = ax^{2} + bx + c\). However, in this function, there is no constant 'c', so it can be assumed as zero. So, rewrite the function as \(y = 6x^{2} - 4x + 0\).

Step 2: Find the vertex of the parabola

The vertex of a parabola \(y = ax^{2} + bx + c\) is given by \((-b/2a, f(-b/2a))\). Here, 'a' equals 6 and 'b' equals -4. Plug these values into the formula to get the vertex. The x-coordinate of the vertex is \(-(-4)/(2*6) = 1/3\), and the y-coordinate is obtained by substituting the x-coordinate into the equation: \(y = 6*(1/3)^{2} - 4*(1/3) = 6/9 - 4/3 = 2/3 - 4/3 = -2/3\). So the vertex is \((1/3, -2/3)\).

Step 3: Sketch the graph

Draw an upward-opening parabola since 'a' is positive. Plot the vertex at \((1/3, -2/3)\). The sketch should show the vertex at \((1/3, -2/3)\) and the parabola opening upwards. Label the vertex in the graph.

Key Concepts

Understanding Quadratic Functions

Quadratic functions are mathematical expressions that describe parabolas, which are U-shaped curves on a two-dimensional plane. At the core of understanding quadratic functions is the standard form equation, which is written as \( y = ax^2 + bx + c \).In this equation, \(x\) and \(y\) are variables that represent the coordinates on the graph, where \(x\) is the independent variable and \(y\) is the dependent variable. The constants \(a\), \(b\), and \(c\) determine the shape and position of the parabola. Here's how they contribute:

• \(a\): This constant affects the parabola's direction (upward for \(a > 0\), downward for \(a < 0\)) and its width (wider for smaller values of \(|a|\), narrower for larger ones).
• \(b\): Influences the position of the vertex relative to the y-axis.
• \(c\):The y-intercept of the parabola, which is where the graph crosses the y-axis when \(x=0\).

When sketching the quadratic function \(y = 6x^2 - 4x\), we noted there is no \(c\) term, meaning the parabola passes through the origin. Understanding this form assists students in identifying the key characteristics needed to sketch the graph properly.

Finding the Vertex of a Parabola

The vertex of a parabola is the tip of the curve --- the highest or lowest point, depending on its orientation. For the upward or downward opening parabolas described by quadratic functions, locating the vertex is crucial for graph sketching. The coordinates of the vertex \((h, k)\) can be calculated using the formula \(h = -b/(2a)\) and \(k = f(h)\).To apply this formula to our function \(y = 6x^2 - 4x\), we first identify the coefficients \(a = 6\) and \(b = -4\). Plugging these into the vertex formula gives us the x-coordinate of the vertex as \(1/3\). To find the y-coordinate, we substitute \(x = 1/3\) back into the original equation, yielding \(-2/3\). Therefore, the vertex of the parabola is located at \((1/3, -2/3)\).

This step is often where students may struggle, but visualizing the function and using the formula methodically can help ensure accuracy. Annotations during calculations can also assist in keeping track of the process for educational purposes.

Sketching Parabola Graphs

Once we have the vertex, we can begin sketching the parabola. Given the function \(y = 6x^2 - 4x\), we know the parabola opens upwards because the coefficient \(a\) is positive. This means our parabola will be U-shaped with the vertex as the lowest point.

We start by plotting the vertex on the graph at \((1/3, -2/3)\). Then, considering the symmetry of parabolas, we plot additional points by selecting x-values around the vertex and calculating the corresponding y-values. Once a few points are plotted, we can draw a smooth curve through them, ensuring to reflect the graph across the axis of symmetry, which is a vertical line running through the vertex.It is also helpful to identify where the graph intersects with the axes. In our function, since there is no \(c\) term, the parabola passes through the origin, which gives us another point to plot.

Exercise Improvement Advice

To help students better grasp this concept, it’s useful to encourage practice with different values of \(a, b,\) and \(c\) to see how each part of the quadratic function influences the shape and position of the graph. Interactive graphing tools can also be incredibly beneficial for visual learners to understand the dynamics of quadratic graphs.