Problem 38 - 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. $$ f(x)=-2 x^{2}+8 x-3 $$

Step-by-Step Solution

Answer

Y-intercept: (0, -3), axis of symmetry: \(x = 2\), vertex: (2, 5). Graph the parabola using these points.

Step 1: Find the y-intercept

The y-intercept occurs when \(x = 0\). Substitute \(x = 0\) into the function \(f(x) = -2x^2 + 8x - 3\). This gives \(f(0) = -2(0)^2 + 8(0) - 3 = -3\). Thus, the y-intercept is \((0, -3)\).

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

The axis of symmetry for a quadratic function of the form \(ax^2 + bx + c\) is given by \(x = -\frac{b}{2a}\). Here, \(a = -2\) and \(b = 8\). Thus, the equation of the axis of symmetry is \(x = -\frac{8}{2(-2)} = 2\). This is also the x-coordinate of the vertex.

Step 3: Find the y-coordinate of the Vertex

Use the x-coordinate of the vertex found in Step 2. Substitute \(x = 2\) into the function: \(f(2) = -2(2)^2 + 8(2) - 3 = -8 + 16 - 3 = 5\). Thus, the vertex is \((2, 5)\).

Step 4: Create a Table of Values

Choose \(x\)-values around the vertex \(x = 2\) to see the behavior of the function. Calculate \(f(x)\) for \(x = 0, 1, 2, 3, 4\):- \(f(0) = -3\)- \(f(1) = 3\)- \(f(2) = 5\)- \(f(3) = 3\)- \(f(4) = -3\).

Step 5: Graph the Function

Using the points from the table \((0, -3), (1, 3), (2, 5), (3, 3), (4, -3)\), plot these on a coordinate plane. Draw a parabola opening downwards since the leading coefficient \(a = -2\) is negative. The vertex \((2, 5)\) is the highest point of the parabola, and the axis of symmetry is the vertical line \(x = 2\).

Key Concepts

Vertex Form

Quadratic functions can be expressed in different forms, and one of these is called the **vertex form**. The vertex form of a quadratic function is written as \[ f(x) = a(x-h)^2 + k \] where

• \( a \) tells us the direction of the parabola (upwards if positive, downwards if negative)
• \( h \) and \( k \) represent the coordinates of the vertex, \((h, k)\)

The vertex is a crucial part of understanding the graph of the quadratic function, as it represents either the highest or lowest point of the parabola.
Converting a quadratic function from standard form \( ax^2 + bx + c \) to vertex form helps in directly identifying the vertex.
Understanding how to identify or convert to vertex form can greatly simplify graphing quadratics.

Axis of Symmetry

The **axis of symmetry** is an important concept when dealing with quadratic functions. It is a vertical line that runs through the vertex of the parabola, and every parabola is symmetric about this line.
For the quadratic function in the form \( ax^2 + bx + c \), the axis of symmetry can be found using the formula \[ x = -\frac{b}{2a} \].
This formula gives us the x-coordinate of the vertex, which lies on the axis of symmetry. Similarly, it helps to locate the peak or trough of the parabola.
Once you find the axis of symmetry, it becomes much easier to plot the parabola since you know the curve will mirror perfectly on each side of this line.

Parabola

A **parabola** is the U-shaped graph generated by a quadratic function. It can open upwards or downwards depending on whether the leading coefficient \( a \) in the quadratic equation is positive or negative.

• If \( a \) is positive, the parabola opens upwards like a smile
• If \( a \) is negative, it opens downwards like a frown

The vertex of the parabola provides a lot of information about the graph:

• If it opens upwards, the vertex is the lowest point (minimum)
• If it opens downwards, the vertex is the highest point (maximum)

Every point on a parabola is equidistant from a point called the "focus" and a line called the "directrix"; however, this is more advanced and not usually needed to sketch basic parabolas.

Intercepts

The **intercepts** of a quadratic function are where the graph crosses the axes. There are two types of intercepts:

• **Y-intercept**: This is where the parabola crosses the y-axis. For any function value at \(x = 0\), substitute \(x\) in the function: \(f(x) = ax^2 + bx + c\) to find the y-intercept, \(c\). For example, in the function \(-2x^2 + 8x - 3\), the y-intercept is \(-3\).
• **X-intercepts**: Also known as roots or zeros, these are the x-values where the parabola crosses the x-axis. Finding these involves solving the quadratic equation \(ax^2 + bx + c = 0\). Depending on the discriminant \(b^2 - 4ac\), there could be 0, 1, or 2 real roots.

Understanding intercepts is key to sketching the position of the parabola on the coordinate plane and providing insights into the nature of the quadratic function.