Problem 14 - Algebra 1: Concepts and Skills

Question

Sketch the graph of the function. Label the coordinates of the vertex. Write an equation for the axis of symmetry. $$ y=2 x^{2}-8 x+3 $$

Step-by-Step Solution

Answer

The coordinates of the vertex are (2, -1). The equation for the axis of symmetry is \(x = 2\).

Step 1: Identify a, b, and c in the quadratic function

For the equation \(y = 2x^2 - 8x + 3\), identify \(a, b, c\) as 2, -8, and 3 respectively.

Step 2: Calculate the vertex

The x-coordinate of the vertex is found using the formula \(-b / (2a)\). Substitute \(a = 2\) and \(b = -8\) into the formula get the x-coordinate of the vertex is \(2\). Substitute \(x = 2\) back into the function to get the y-coordinate, \(y = 2*(2)^2 - 8*(2) + 3 = -1\). Therefore, the vertex is \((2, -1)\).

Step 3: Calculate the axis of symmetry

The equation for the axis of symmetry is \(x=h\), where \(h\) is the x-coordinate of the vertex. Substituting \(h = 2\) gives the equation for the axis of symmetry as \(x=2\).

Step 4: Sketch the graph

Plot the vertex and the axis of symmetry on the graph. Draw a symmetrical parabola that opens upwards (since \(a>0\)) with the axis of symmetry as the line of symmetry. Label the vertex and axis of symmetry on the graph.

Key Concepts

Vertex Calculation

The vertex of a quadratic function is a crucial point that helps define the shape and position of its graph. In the quadratic equation \(y = ax^2 + bx + c\), the vertex can be calculated using a straightforward formula. The x-coordinate of the vertex is given by \(-\frac{b}{2a}\). This formula allows you to find the point where the parabola either reaches its maximum or minimum value, depending on the concavity.

For instance, in the function \(y = 2x^2 - 8x + 3\), identify that \(a = 2\) and \(b = -8\). Plug these values into the formula to find the x-coordinate:

• \(-\frac{-8}{2 \times 2} = 2\)

With the x-coordinate as \(2\), substitute back into the function to find the y-coordinate:

• \(y = 2(2)^2 - 8(2) + 3 = -1\)

This gives the vertex at the point \((2, -1)\). The vertex not only serves as the turning point of the parabola but is also used to plot the graph accurately.

Axis of Symmetry

The axis of symmetry of a quadratic function is an imaginary line that divides the parabola into two symmetrical halves. It's essential for understanding how the graph is structured because it gives the line around which the parabola is mirrored.

For any quadratic function \(y = ax^2 + bx + c\), the equation for the axis of symmetry can be found using the vertex x-coordinate derived from \(-\frac{b}{2a}\).

In the example \(y = 2x^2 - 8x + 3\), we determined the x-coordinate of the vertex to be \(2\). Therefore, the axis of symmetry is simply the line \(x = 2\). This line cuts the parabola perfectly in half and helps in sketching an accurate graph. It is essential for ensuring that the two sides of the parabola reflect each other exactly.

Parabola Sketching

Sketching a parabola requires understanding the direction in which it opens and using key points like the vertex and the axis of symmetry. A quadratic function will have a parabolic graph, either opening upwards if \(a > 0\), or downwards if \(a < 0\).

For the function \(y = 2x^2 - 8x + 3\), because \(a = 2\), the parabola opens upwards. Here are the core steps to sketch:

• Plot the vertex \((2, -1)\).
• Draw the axis of symmetry \(x = 2\).
• Mark a couple of additional points by choosing x-values around the vertex, calculating corresponding y-values, and ensuring equal distances from the axis for a symmetrical look.
• Connect these points smoothly to form the U-shape of the parabola.

Label the vertex and the axis of symmetry on the graph as these guide the structure of your parabola. Understanding these elements makes drawing the parabola more intuitive and accurate.