Problem 37 - 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. $$ f(x)=3 x^{2}+6 x-1 $$

Step-by-Step Solution

Answer

y-intercept: (0, -1), axis of symmetry: x = -1, vertex: (-1, -4).

Step 1: Finding the y-intercept

To find the y-intercept of the quadratic function, substitute \( x = 0 \) into the function \( f(x) = 3x^2 + 6x - 1 \).\[ f(0) = 3(0)^2 + 6(0) - 1 = -1 \]Therefore, the y-intercept is \( (0, -1) \).

Step 2: Finding the Axis of Symmetry and x-coordinate of Vertex

The equation of the axis of symmetry for a quadratic function \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \).Substitute \( a = 3 \) and \( b = 6 \) into the formula:\[ x = -\frac{6}{2(3)} = -1 \]Hence, the axis of symmetry is \( x = -1 \), and the vertex's x-coordinate is \( -1 \).

Step 3: Finding the y-coordinate of the Vertex

To find the y-coordinate of the vertex, substitute \( x = -1 \) into the function \( f(x) = 3x^2 + 6x - 1 \).\[ f(-1) = 3(-1)^2 + 6(-1) - 1 = 3 - 6 - 1 = -4 \]Thus, the vertex is \( (-1, -4) \).

Step 4: Creating a Table of Values

Include the vertex and several points on either side of the vertex. Here is a possible table of values:| \( x \) | \( f(x) \) ||-----|----------|| -3 | 14 || -2 | 3 || -1 | -4 || 0 | -1 || 1 | 8 |

Step 5: Graphing the Function

Use the points in the table and the y-intercept to sketch the parabola. Plot the points \((-3, 14), (-2, 3), (-1, -4), (0, -1), (1, 8)\) and draw a smooth curve through them, noting that the axis of symmetry is at \( x = -1 \) and the parabola opens upwards.

Key Concepts

Understanding the y-intercept in Quadratic Functions

In any function, including quadratic functions, the y-intercept gives a clear starting point: it is where the graph crosses the y-axis. For a quadratic function like \( f(x) = 3x^2 + 6x - 1 \), you can find this point by setting \( x = 0 \) and solving for \( f(x) \). This is because, when \( x = 0 \), the x-term disappears, leaving the y-intercept as a direct outcome of the constant and squared terms.
In our example:

• Replace \( x \) with 0 in the equation: \( 3(0)^2 + 6(0) - 1 = -1 \).
• The y-intercept is \( (0, -1) \).

This tells us that the point where the graph of the function will meet the y-axis is at \( (0, -1) \).
Knowing the y-intercept helps in sketching the graph because it's one of the key points you can plot right away. Every graph of a quadratic equation will have one y-intercept unless modified by a transformation that affects where the graph crosses the y-axis.

The Axis of Symmetry: Balancing the Parabola

The axis of symmetry is an essential feature of quadratic functions. It gives the line that the parabola is perfectly mirrored across. For any quadratic function \( ax^2 + bx + c \), the axis of symmetry is found using the formula \( x = -\frac{b}{2a} \). This axis runs vertically through the vertex and splits the parabola into two symmetrical halves, meaning one side of the graph reflects the other.
Let's work it out for our function:

• Identify \( a = 3 \) and \( b = 6 \).
• Substitute these into the formula: \( x = -\frac{6}{2 \times 3} = -1 \).

Therefore, the axis of symmetry for this function is at \( x = -1 \).
This knowledge is crucial when plotting the graph because it tells us where the peak or trough of the parabola is situated. Without this, the symmetry of the graph would be difficult to capture accurately.

Finding the Vertex of a Parabola

The vertex of a parabola is a significant point which shows the highest or lowest point of the graph, depending on its orientation. For a quadratic function, the vertex acts as a guide point for shaping the parabola. It is located on the axis of symmetry, giving its position a central role in the curvature of the graph.
In the function \( f(x) = 3x^2 + 6x - 1 \), you have already found the x-coordinate of the vertex as -1 because it lies on the axis of symmetry. Now, you use this to find the y-coordinate:

• Substitute \( x = -1 \) back into the function to get \( f(-1) \).
• Calculate: \( f(-1) = 3(-1)^2 + 6(-1) - 1 = -4 \).

This gives a vertex of \( (-1, -4) \).
The vertex is incredibly useful when graphing the quadratic because it indicates the maximum or minimum value of the function. For our example, because the coefficient of \( x^2 \) is positive, the parabola opens upwards, and the vertex \( (-1, -4) \) is its lowest point. This is a critical visual clue as you draw the graph.