Problem 52 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The graph of the function \(y = 3x^2 - 2x\) is a parabola opening upwards with the vertex at \((1/3, -1/3)\). The axis of symmetry is \(x = 1/3\).

Step 1: Identify the Coefficients

Determine the values of \(a\), \(b\), and \(c\) from function \(y = 3x^2 - 2x\). Here, \(a = 3\), \(b = -2\), and \(c = 0\).

Step 2: Calculate the Vertex

The formula to determine the x-coordinate of the vertex is \(x = -b/2a = -(-2)/2*3 = 1/3\). Substitute \(x = 1/3\) in the equation to find y-coordinate. Thus, \(y = 3(1/3)^2 - 2*(1/3) = 1/3 - 2/3 = -1/3\). Therefore, the vertex is \((1/3, -1/3)\).

Step 3: Draw the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex. The equation for the axis of symmetry is \(x = -b/2a\). So, \(x = 1/3\).

Step 4: Sketch the Parabola

Draw a graph with the vertex at \((1/3, -1/3)\), which is the lowest point as this graph opens upwards because \(a\) is positive. The graph is symmetrical with respect to the line \(x = 1/3\). Draw the graph accordingly.

Key Concepts

Vertex of a Parabola

Understanding the vertex of a parabola is crucial for graphing quadratic functions accurately. The vertex represents the highest or lowest point on a parabola, depending on its direction. For the function \( y = ax^2 + bx + c \) with a positive \( a \) value, the parabola opens upwards and the vertex is the minimum point. Conversely, with a negative \( a \) value, it opens downwards and the vertex is the maximum point.

To find the vertex, you can use the formula \( x = -\frac{b}{2a} \) for the x-coordinate. Then, substitute this value back into the original equation to find the corresponding y-coordinate. For the given function \( y = 3x^2 - 2x \) from the exercise, we calculated the x-coordinate of the vertex as \( 1/3 \) and then determined the y-coordinate to be \( -1/3 \) by substitution, giving us the vertex \( (1/3, -1/3) \).

Hence, plotting the vertex correctly is the first key step in sketching the graph of the quadratic function.

Axis of Symmetry

The axis of symmetry in a quadratic function graph is a vertical line that divides the graph into two mirror images. For all parabolas, the axis passes through the vertex. The general equation for the axis of symmetry is \( x = -\frac{b}{2a} \), derived from the vertex formula.

For our example \( y = 3x^2 - 2x \), we've already established that the x-coordinate of the vertex is \( 1/3 \), which is also the equation of the axis of symmetry. This means the parabola is symmetrical around the line \( x = 1/3 \).

When sketching a quadratic graph, marking the axis of symmetry provides a reference that ensures the two halves of the parabola are congruent. Visually checking this symmetry can also help you verify the accuracy of your graph.

Parabola Graph

The graph of a quadratic function, or a parabola graph, represents the set of all points \( (x, y) \) that satisfy the function \( y = ax^2 + bx + c \). Sketching the parabola efficiently involves plotting the vertex, drawing the axis of symmetry, and then plotting additional points on either side to show the curve of the graph.

In the case of \( y = 3x^2 - 2x \), after placing the vertex and drawing the axis of symmetry, we recognize that since \( a > 0 \) the parabola opens upwards. It's helpful to list values around the vertex, often selecting x-values equidistant from the axis of symmetry so that you can take advantage of the symmetry when plotting points.

After plotting, you would then draw a smooth curve through these points, ensuring it passes through the vertex and touches the x-axis at any real roots, if applicable. The result is a symmetrical parabola graph, which showcases the fundamental characteristics of quadratic functions.