Problem 28 - 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. $$ y=5 x^{2}-x $$

Step-by-Step Solution

Answer

The graph of the function opens upwards. The vertex is at (1/10, 0.45). The equation of the axis of symmetry for the function is x = 1/10.

Step 1: Determine the opening of the graph

The coefficient of \(x^2\) in our function is \(5\), which is a positive number. Therefore, the graph of the function opens upwards.

Step 2: Finding the vertex

The x-coordinate of the vertex can be found using \(x = -b/2a\). Here, \(a = 5\) and \(b = -1\), thus the x-coordinate of the vertex is \(-(-1) / (2*5) = 1/10\). To find the y-coordinate, we substitute \(x = 1/10\) into our function to get \(y = 5*(1/10)^{2} - (1/10) = 0.45\). So, the vertex is \((1/10, 0.45)\).

Step 3: Writing the equation of the axis of symmetry

The axis of symmetry is simply the vertical line through the vertex, in other words, it's the x-coordinate of the vertex. Thus, the equation for the axis of symmetry is \(x = 1/10\).

Key Concepts

Parabola

A parabola is a U-shaped curve on a graph, and it represents the graphical solution of a quadratic equation. When dealing with quadratic functions of the form \(y = ax^2 + bx + c\), the value of \(a\) determines how the parabola opens. If \(a > 0\), the parabola opens upwards. Conversely, if \(a < 0\), it opens downwards. This behavior occurs because the sign of \(a\) dictates the direction in which the function increases or decreases.

• Open up (\(a>0\)) - The parabola looks like a U.
• Open down (\(a<0\)) - The parabola looks like an upside-down U.

When tackling problems involving quadratic functions, determining whether the parabola opens up or down is a crucial first step, as it impacts the rest of the graph's characteristics.

Vertex

The vertex of a parabola is its peak, the highest or lowest point on the graph depending on the direction the parabola opens. It gives us vital information about the function's maximum or minimum value. Finding the vertex helps in understanding the graph's symmetry and its highest or lowest point.
For a quadratic function \(y = ax^2 + bx + c\), you can find the vertex using the formula for the x-coordinate, \(x = -b/(2a)\). By substituting this x-value back into the function, you can find the corresponding y-coordinate. This results in the vertex coordinates.

• x-coordinate: \(x = -\frac{b}{2a}\)
• Substitute into the function to find the y-coordinate

For the example function \(y = 5x^2 - x\), with \(a=5\) and \(b=-1\), the vertex is calculated as \((\frac{1}{10}, 0.45)\). This point indicates the minimum of this upward-opening parabola.

Axis of Symmetry

The axis of symmetry in a quadratic function is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex and will always be a line parallel to the y-axis. Understanding the axis of symmetry helps simplify calculations and graph sketches by providing a central reference for reflecting points across the parabola.
For any quadratic function, the equation for the axis of symmetry can be found using the x-coordinate of the vertex, since the axis runs along this vertical line.
In the function \(y = 5x^2 - x\), the vertex's x-coordinate is \(\frac{1}{10}\). Hence, the equation for the axis of symmetry is \(x = \frac{1}{10}\). This line acts as a mirror, ensuring that for every point on one side, there is an equidistant point on the other.