Problem 36 - Algebra 1

Question

Complete these steps for the function. a. Tell whether the graph of the function opens up or down. b. Find the coordinates of the vertex. c. Write an equation of the axis of symmetry. $$ y=2 x^{2}+7 x-21 $$

Step-by-Step Solution

Answer

a. The graph opens upwards because the coefficient of \(x^2\) is positive. b. The vertex of the graph is at \((-7/4, -31/8)\). c. The equation of the axis of symmetry is \(x=-7/4\).

Step 1: Determining the direction of the parabola

The given function is \(y=2x^2+7x-21\). The coefficient of the \(x^2\) term, 2, is positive. Thus, the graph of the function opens upwards.

Step 2: Finding the vertex

Using the formula, the x-coordinate of the vertex is given by \(-b/2a=-7/(2*2)=-7/4\). Substituting \(-7/4\) into the equation gives us the y-coordinate of the vertex: \(y=2(-7/4)^2+7(-7/4)-21=-31/8\). Therefore, the vertex of the function is \((-7/4,-31/8)\).

Step 3: Writing the equation of the axis of symmetry

The equation of the axis of symmetry is given by the line \(x=h\), where h is the x-coordinate of the vertex. Thus, the equation of the axis of symmetry of this function is \(x=-7/4\).

Key Concepts

Parabola

A parabola is a U-shaped curve that you often encounter in quadratic functions. In our example, the quadratic function is given by the equation \(y = 2x^2 + 7x - 21\). This equation represents a parabola. The general form of a quadratic equation is \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

• If the coefficient \(a\) is positive, like in our case where \(a = 2\), the parabola opens upwards.
• If \(a\) is negative, the parabola would open downwards.

Understanding whether a parabola opens upward or downward can help you easily determine what's happening to the graph as \(x\) changes.

Vertex

The vertex of a parabola is a crucial point. It's either the highest or lowest point on the graph, depending on the direction the parabola opens.
In the quadratic equation \(y = 2x^2 + 7x - 21\), we can find the vertex by using the formula \(x = -b/2a\). For our function:

• The \(x\)-coordinate of the vertex is \(-b/2a = -7/(2 \times 2) = -7/4\).
• To find the \(y\)-coordinate, substitute \(x = -7/4\) back into the equation: \(y = 2(-7/4)^2 + 7(-7/4) - 21 = -31/8\).
• Hence, the vertex is at \((-7/4, -31/8)\).

For a parabola that opens upwards, like ours does, the vertex represents the lowest point on the graph.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It essentially slices the parabola right in the middle at the vertex.
For any quadratic function, the axis of symmetry can be easily identified as the line \(x = h\), where \(h\) is the \(x\)-coordinate of the vertex.

• In our function \(y = 2x^2 + 7x - 21\), the \(x\)-coordinate of the vertex is \(-7/4\).
• Therefore, the equation of the axis of symmetry is \(x = -7/4\).

This imaginary line is crucial for graphing, as it often helps in sketching the parabola accurately and understanding its symmetry.