Problem 53 - Algebra 2

Question

Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x-\) -coordinate of the vertex for each quadratic function. Then graph the function by making a table of values. $$ f(x)=-4 x^{2}+8 x-1 $$

Step-by-Step Solution

Answer

Y-intercept: (0, -1); Axis of symmetry: x = 1; X-coordinate of vertex: 1.

Step 1: Identify the Quadratic Function

The given function is a quadratic function of the form \( f(x) = ax^2 + bx + c \), where \(a = -4\), \(b = 8\), and \(c = -1\). We will use the properties of this function to find the requested characteristics.

Step 2: Find the Y-intercept

The y-intercept of a function is the value of \(f(x)\) when \(x = 0\). Substitute \(x = 0\) into the function: \[ f(0) = -4(0)^2 + 8(0) - 1 = -1 \]. Thus, the y-intercept is \((0, -1)\).

Step 3: Determine the Axis of Symmetry

The axis of symmetry for a quadratic function \(ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Using \(b = 8\) and \(a = -4\), the axis of symmetry is \[ x = -\frac{8}{2(-4)} = 1 \].

Step 4: Calculate the X-coordinate of the Vertex

The x-coordinate of the vertex is the same as the x-value of the axis of symmetry. Therefore, the x-coordinate is \( x = 1 \).

Step 5: Make a Table of Values

To graph the function, choose values of \(x\) around the vertex and calculate corresponding \(f(x)\) values. For example, choose \(x = 0, 1, 2\):- \(f(0) = -1\)- \(f(1) = 3\) (vertex)- \(f(2) = -1\)

Step 6: Graph the Function

Plot the points found in the table: \((0, -1)\), \((1, 3)\), and \((2, -1)\). Draw the parabola that passes through these points, indicating the axis of symmetry at \(x = 1\).

Key Concepts

Understanding the Y-intercept

In a quadratic function, the y-intercept is an important feature because it tells us where the graph of the function will intersect the y-axis. To find the y-intercept of a quadratic function, you substitute 0 for the value of x in the function equation:

• For the function given, \( f(x) = -4x^2 + 8x - 1 \), the y-intercept is calculated as \( f(0) = -4(0)^2 + 8(0) - 1 = -1 \).
• This means that the graph will cross the y-axis at the point (0, -1).

The y-intercept helps in graphing as it gives you one of the starting points to sketch the parabola.
The coordinate (0, -1) is the specific point on the graph where the curve touches the y-axis.

Axis of Symmetry in Quadratic Functions

The axis of symmetry is a vertical line that runs through the vertex of a parabola. It essentially divides the parabola into two mirror-image halves.To find the axis of symmetry for a quadratic function expressed as \( ax^2 + bx + c \), use the formula: \[ x = -\frac{b}{2a} \] Here's how it works with our function, \( f(x) = -4x^2 + 8x - 1 \):

• Given \( a = -4 \) and \( b = 8 \), substitute these into the axis of symmetry formula: \( x = -\frac{8}{2(-4)} = 1 \).

This tells us that the axis of symmetry is the vertical line \( x = 1 \).
This line is crucial when graphing because it helps visualize the parabola's symmetry and ensures your graph's accuracy.

Identifying the Vertex of a Parabola

The vertex is one of the most significant points on a parabola, either representing the highest or lowest point, depending on the parabola's orientation.In a quadratic function, the x-coordinate of the vertex is the same as the x-value of the axis of symmetry.

• In \( f(x) = -4x^2 + 8x - 1 \), since the axis of symmetry is \( x = 1 \), the vertex's x-coordinate is also 1.
• To find the y-coordinate, substitute \( x = 1 \) into the function: \( f(1) = -4(1)^2 + 8(1) - 1 = 3 \).

Thus, the vertex of the function is at the point (1, 3).
Understanding the vertex is vital as it provides insight into the direction and position of the parabola on the graph.

Graphing Quadratic Functions

Graphing a quadratic function involves understanding its shape and key features like the vertex, axis of symmetry, and y-intercept.Here’s how you can graph \( f(x) = -4x^2 + 8x - 1 \):

• First, plot the y-intercept from (0, -1) on the graph.
• Next, mark the vertex at (1, 3) which is also a crucial turning point of the parabola.
• Use the axis of symmetry, \( x = 1 \), to ensure the graph is symmetrical on either side of the vertex.
• Create a table with x-values around the vertex, such as x = 0, 1, 2. This gives you more points to define the parabola: \( f(0) = -1 \), \( f(1) = 3 \), and \( f(2) = -1 \).

Finally, connect these points with a smooth curved line to sketch the parabola.
With these steps, you’ll have a complete graph of the quadratic function, showcasing its symmetrical parabolic curve.