Problem 6 - Algebra 2

Question

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=3 x^{2}+10 x $$

Step-by-Step Solution

Answer

y-intercept: 0; axis of symmetry: \(x = -\frac{5}{3}\); vertex: \((-\frac{5}{3}, -\frac{25}{3})\). Graph is a parabola opening upwards.

Step 1: Find the y-intercept

To find the y-intercept of the quadratic function, substitute \(x = 0\) into the equation \(f(x) = 3x^2 + 10x\). This gives us \(f(0) = 3(0)^2 + 10(0) = 0\). Thus, the y-intercept is \(0\).

Step 2: Determine the Equation of the Axis of Symmetry

The equation for the axis of symmetry of a quadratic function \(ax^2 + bx + c\) is given by \(x = -\frac{b}{2a}\). For the given function, \(a = 3\) and \(b = 10\). Substituting, \(x = -\frac{10}{2 imes 3} = -\frac{10}{6} = -\frac{5}{3}\). Thus, the axis of symmetry is \(x = -\frac{5}{3}\).

Step 3: Find the x-coordinate of the Vertex

The x-coordinate of the vertex is the same as the axis of symmetry: \(x = -\frac{5}{3}\).

Step 4: Calculate the y-coordinate of the Vertex

Substitute the x-coordinate of the vertex \(-\frac{5}{3}\) into the function to find the y-coordinate: \[f\left(-\frac{5}{3}\right) = 3\left(-\frac{5}{3}\right)^2 + 10\left(-\frac{5}{3}\right) = 3\times\frac{25}{9} - \frac{50}{3}\] Solving this gives:\[= \frac{75}{9} - \frac{150}{9} = \frac{-75}{9} = -\frac{25}{3}\] Thus, the vertex is \((-\frac{5}{3}, -\frac{25}{3})\).

Step 5: Create a Table of Values

Choose values of \(x\) around the vertex \(x = -\frac{5}{3}\). Calculate the corresponding \(f(x)\) for \(x = -2\), \(-1\), \(0\), and \(-3\):- \(f(-3) = 3(-3)^2 + 10(-3) = 27 - 30 = -3\)- \(f(-2) = 3(-2)^2 + 10(-2) = 12 - 20 = -8\)- \(f(-1) = 3(-1)^2 + 10(-1) = 3 - 10 = -7\)- \(f(0) = 0\)| \(x\) | \(f(x)\) ||-------|---------|| -3 | -3 || -2 | -8 || -1 | -7 || 0 | 0 || -5/3 | -25/3 |

Step 6: Graph the Function

To graph the function, plot the points from the table of values and the vertex \((-\frac{5}{3}, -\frac{25}{3})\). Draw the axis of symmetry at \(x = -\frac{5}{3}\). Sketch the parabola opening upwards as the leading coefficient \(3 > 0\). Make sure it fits through all points, maintaining symmetry relative to the line \(x = -\frac{5}{3}\).

Key Concepts

Y-Intercept

The y-intercept of a quadratic function is where the graph intersects the y-axis. This point occurs when the x-value is zero. To find the y-intercept, substitute \(x = 0\) into the quadratic equation. For the function \(f(x) = 3x^2 + 10x\), it becomes:

• \(f(0) = 3(0)^2 + 10(0) = 0\)

Thus, the y-intercept is at the point (0, 0), meaning the graph of the quadratic function crosses the y-axis at this point. Understanding the y-intercept is essential for drawing the initial framework of the graph.

Axis of Symmetry

In the graph of a quadratic function, the axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves. The equation for the axis of symmetry can be calculated using the formula:\[x = -\frac{b}{2a}\]where \(a\) and \(b\) are coefficients from the function \(ax^2 + bx + c\). For our function, \(a = 3\) and \(b = 10\), resulting in:

• \(x = -\frac{10}{6} = -\frac{5}{3}\)

The axis of symmetry here is \(x = -\frac{5}{3}\). This line is crucial as it helps in identifying the vertex and ensures that our parabola is accurately plotted with symmetry.

Vertex

The vertex of a parabola is its highest or lowest point, depending on whether it opens upwards or downwards. For quadratic functions opening upwards, like \(f(x) = 3x^2 + 10x\), the vertex is the minimum point. The x-coordinate of the vertex is the same as the axis of symmetry, \(-\frac{5}{3}\). To find the y-coordinate, plug this x-value into the function:\[f\left(-\frac{5}{3}\right) = 3\left(-\frac{5}{3}\right)^2 + 10\left(-\frac{5}{3}\right) = -\frac{25}{3}\]Thus, the vertex of the graph is at \((-\frac{5}{3}, -\frac{25}{3})\). Knowing the vertex is key because it represents the tip of the parabola, giving insight into the graph's shape and direction.

Parabola

A parabola is the shape of the graph of a quadratic function. It has a curved, symmetrical arc that can open either upwards or downwards:

• If the leading coefficient (in front of \(x^2\)) is positive, like 3 in \(f(x) = 3x^2 + 10x\), the parabola opens upwards.
• If it is negative, the parabola opens downwards.

The width, orientation, and position of the parabola are determined by the coefficients in the quadratic function and the locations of the vertex and axis of symmetry. Understanding the basic properties of a parabola helps to predict how it behaves on a graph.

Graphing Quadratics

Graphing quadratic functions involves plotting the y-intercept, vertex, and additional points to sketch the parabola. Begin by identifying the intercept, axis of symmetry, and vertex coordinates to set key points. Use a table of values to determine other points through which the graph should pass. For example:

• Use points such as \((-3, -3)\), \((-2, -8)\), \((-1, -7)\), and \((0, 0)\) as determined in the calculation process.
• Draw the axis of symmetry (e.g., \(x = -\frac{5}{3}\)) as a dashed line for reference.
• Sketch the parabola, ensuring symmetry about this axis and that the graph passes through all plotted points.

Graphing a quadratic function visually represents its behavior and helps in understanding its properties such as direction, symmetry, and vertex position.