Exercise 10 Page 534

We are asked to solve the given quadratic equation by graphing. There are three steps to solving a quadratic equation by graphing.

1. Write the equation in standard form, ax^2+bx+c=0.
2. Graph the related function y=ax^2+bx+c.
3. Find the x-intercepts, if any.

The solutions of ax^2+bx+c=0 are the x-intercepts of the graph of y=ax^2+bx+c. Let's write our equation in standard form. This means gathering all of the terms on the left-hand side of the equation. x^2-2x=- 4 ⇔ x^2-2x+4=0 Now we can identify the function related to the equation. Equation:&x^2-2x+4=0 Related Function:&y=x^2-2x+4

Graphing the Related Function

To draw the graph of the related function written in standard form, we must start by identifying the values of a, b, and c. y=x^2-2x+4 ⇔ y=1x^2+(- 2)x+4 We can see that a=1, b=- 2, and c=4. Now, we will follow four steps to graph the function.

1. Find the axis of symmetry.
2. Calculate the vertex.
3. Identify the y-intercept and its reflection across the axis of symmetry.
4. Connect the points with a parabola.

Finding the Axis of Symmetry

The axis of symmetry is a vertical line with equation x=- b2a. Since we already know the values of a and b, we can substitute them into the formula.

The axis of symmetry of the parabola is the vertical line with equation x=1.

Calculating the Vertex

To calculate the vertex, we need to think of y as a function of x, y=f(x). We can write the expression for the vertex by stating the x- and y-coordinates in terms of a and b. Vertex: ( - b/2a, f( - b/2a ) ) Note that the formula for the x-coordinate is the same as the formula for the axis of symmetry, which is x=1. Thus, the x-coordinate of the vertex is also 1. To find the y-coordinate, we need to substitute 1 for x in the given equation.

We found the y-coordinate, and now we know that the vertex is (1,3).

Identifying the y-intercept and its Reflection

The y-intercept of the graph of a quadratic function written in standard form is given by the value of c. Thus, the point where our graph intercepts the y-axis is (0,4). Let's plot this point and its reflection across the axis of symmetry.

Connecting the Points

We can now draw the graph of the function. Since a=1, which is positive, the parabola will open upwards. Let's connect the three points with a smooth curve.

Finding the x-intercepts

Let's identify the x-intercepts of the graph of the related function, if there are any.

We can see that the parabola does not intercept the x-axis. Therefore, the equation x^2-2x=- 4 has no solutions.