Problem 34 - 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. $$ y=-4 x^{2}-4 x+8 $$

Step-by-Step Solution

Answer

The function opens downward. The vertex is at (-1/2, 9). The equation of the axis of symmetry is x=-1/2.

Step 1: Determine Direction

A quadratic function \(y=ax^{2}+bx+c\) opens upward if \(a>0\) and downward if \(a<0\). Given \(y=-4x^{2}-4x+8\), the coefficient for \(x^{2}\) is -4, which is less than 0. Hence, the function opens downward.

Step 2: Find the Vertex

The vertex of a quadratic function \(y=ax^{2}+bx+c\) is given by \((-b/2a , f(-b/2a))\). For the function, the coefficients \(a=-4\) and \(b=-4\). Plugging these into the formula, the \(x\)-coordinate of the vertex is \(-(-4)/2(-4) = -1/2\). Substituting \(-1/2\) into the given function, we get \(y=-4(-1/2)^{2}-4(-1/2)+8 = -1+2+8 = 9\). Thus, the vertex of the function is \((-1/2, 9)\).

Step 3: Write the Equation of the Axis of Symmetry

The axis of symmetry of a quadratic function \(y=ax^{2}+bx+c\) is a vertical line passing through the vertex, and its equation is given by \(x = -b/2a\). This equation will give the same \(x\)-coordinate as found in Step 2. Thus, the equation of the axis of symmetry is \(x = -1/2\).

Key Concepts

Vertex

The vertex of a quadratic function is a critical point on the graph of a parabola. It is where the curve turns around, and represents the highest or lowest point depending on the graph's direction. In general, the vertex can be found using the formula:

• The x-coordinate is \(-\frac{b}{2a}\).
• The y-coordinate is found by plugging this x value back into the equation.

For the quadratic equation \(y=-4x^2-4x+8\), the vertex is calculated as follows:

• Calculate \(-\frac{-4}{2\times(-4)} = -\frac{1}{2}\) for the x-coordinate.
• Substitute \(-\frac{1}{2}\) back into the equation to find the y-coordinate, resulting in \(9\).

So, the vertex is \((-\frac{1}{2}, 9)\). This tells you where the parabola reaches its maximum point since the graph opens downward.

Axis of Symmetry

A parabola's axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For any quadratic equation in the form \(y = ax^2 + bx + c\), the axis of symmetry can be found using the formula:

• \(x = -\frac{b}{2a}\)

Using the coefficients from our equation \(y=-4x^2-4x+8\):

• The axis of symmetry is \(x = -\frac{-4}{2\times(-4)} = -\frac{1}{2}\).

This line passes through the vertex and is key in understanding how the parabola is oriented on the graph. It helps us predict the behavior of the graph and understand its symmetry.

Graph Direction

The direction in which a quadratic graph opens is determined by the sign of the coefficient \(a\) from the quadratic equation \(y = ax^2 + bx + c\). Here's how to determine it:

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), the parabola opens downwards.

In the equation \(y = -4x^2 - 4x + 8\), the coefficient \(-4\) is negative. Therefore, the graph opens downward. This means that the vertex is the highest point on the graph, creating an upside-down U-shape, also known as an inverted parabola.

Quadratic Equation

Quadratic equations are polynomial equations of the second degree, typically expressed as \(y = ax^2 + bx + c\). The values of \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). These equations graph as parabolas on a coordinate plane.A few characteristics of quadratic equations include:

• The highest power of the variable \(x\) is 2.
• The graph is a symmetric curve called a parabola.
• It can have two, one, or no real roots depending on the discriminant \(b^2 - 4ac\).

Quadratic functions are widely used in various fields such as physics, engineering, and business for modeling scenarios that exhibit parabolic behavior, like projectiles, lenses, or profit maximization.

Parabola

A parabola is a U-shaped curve that is the graph of a quadratic function. The shape's defining characteristic is its symmetry, as each side mirrors the other across the axis of symmetry. Parabolas can open upwards or downwards depending on the coefficient of the squared term in the quadratic equation.Key features of a parabola include:

• The vertex, which is the peak or the trough of the curve.
• The axis of symmetry, which is the line dividing the parabola into two equal halves.
• An opening direction determined by the sign of \(a\) in the equation \(y=ax^2+bx+c\).

Parabolas have practical applications in various fields, such as in satellite dishes and car headlights where reflective properties are advantageous, and in physics where projectile motion is modeled.