Question

Quadratic Functions in Standard FormCaroline wrote these steps to graph f(x) = 2x2 + 4x + 5 on note cards, but they gotmixed up. Help Caroline by re-writing the steps in the correct order. Use the notecards to complete the steps below.

Answer 1

The correct steps for graphing the function
`\( f(x) = 2x^2 + 4x + 5 \)` are as follows:

1. The axis of symmetry is at x = -1 . We draw a dashed red line to represent the axis of symmetry on the graph.

2. The vertex of the parabola, found by substituting the x-coordinate of the vertex into the equation, is at (-1, 3). It's marked with a blue point on the graph.

3. The y-intercept is at 5, corresponding to the point (0, 5), marked with a green point on the graph.

4. A symmetrical point over the axis of symmetry to the y-intercept is (-2, 5), which is marked with a purple point on the graph.

The image you've uploaded contains steps for graphing the function
`\( f(x) = 2x^2 + 4x + 5 \)` on note cards, but they are mixed up. To help Caroline, we should put these steps in the correct order and then perform them to graph the function.

Here are the correct steps to graph
`\( f(x) = 2x^2 + 4x + 5 \)`:

1. Find the axis of symmetry: The axis of symmetry for a quadratic function
`\( ax^2 + bx + c \) is given by \( x = -(b)/(2a) \)`. For the given function, a = 2 and b = 4 , so the axis of symmetry is at
`\( x = -(4)/(2 \cdot 2) = -1 \).`

2. Determine the vertex: The vertex lies on the axis of symmetry. To find the y-coordinate of the vertex, substitute x = -1 into the equation. So we calculate f(-1) .

3. Find the y-intercept: This is the point where x = 0 For the given function,( f(0) = 5 so the y-intercept is at (0, 5).

4. Plot a point symmetrical over the axis of symmetry to the y-intercept: Because the axis of symmetry is at x = -1 and the y-intercept is at (0, 5), we'll find a point symmetrical to (0, 5) across x = -1 . This will be two units to the left of x = -1 , at x = -2 .

5. Plot another point and its reflection across the axis of symmetry: Choose another x-value, find its corresponding y-value, and then plot a point symmetrical to it across the axis of symmetry.

6. Draw the parabola: Use the plotted points and the axis of symmetry to sketch the parabola.

Let's calculate the vertex of the parabola and the symmetrical point to the y-intercept. After that, we can sketch a rough graph based on these calculations.

Answer 2

1.- Determine the vertex...

2.- The axis of symmetry...

3.- The y-intercept...

4.- Plot a point...

5.- Plot another point...

The above is the correct order of the cards, now the reason for that first you have to find the vertex. Now, the axis of symmetry is a straight line that passes through the vertex. Later you have to find another 2 points reflected by the symmetry axis and finally construct the graph.