Exercise 34 Page 218

We want to draw the graph of the given quadratic function. To do so, we will rewrite it in vertex form, f(x)=a(x-h)^2+k, where a, h, and k are either positive or negative numbers. To draw the graph, we will follow four steps.

1. Identify the constants a, h, and k.
2. Plot the vertex (h,k) and draw the axis of symmetry x=h.
3. Plot any point on the curve and its reflection across the axis of symmetry.
4. Sketch the curve.

Let's get started.

Step 1

We have a quadratic function written in standard form, and we want to rewrite it in vertex form. Standard Form:& y= ax^2+ bx+ c Given Equation:& y= 2x^2+ 12x- 8Notice that in the standard form equation all of the coefficients are positive. This means that the 8 for c is a - 8. In the given equation, a= 2, b= 12, and c= -8. Let's now recall the vertex form of a quadratic function. Vertex Form: y=a(x-h)^2+k In this equation, a is the leading coefficient of the quadratic function and the point (h,k) is the vertex of the parabola. By substituting our given values for a and b into the expression - b2a, we can find h.

So far, we know that the vertex lies at (-3,k). To find the y-coordinate k, we will substitute -3 for x in the given function.

Therefore, the (h,k) coordinate pair of the vertex is ( -3, -26). Recall that if a<0, the parabola will open downwards. Conversely, if a>0, the parabola will open upwards. Since we know that a= 2, h= -3, and k= -26 we can write it in vertex form. Since a is greater than 0, the parabola will open upwards. Vertex Form:& f(x)= a(x- h)^2+ k Function:& f(x)= 2(x+ 3)^2- 26

Step 2

Let's now plot the vertex ( h, k) and draw the axis of symmetry x= h. Since we already know the values of h and k, we know that the vertex is ( -3, -26). Therefore, the axis of symmetry is the vertical line x= -3.

Step 3

We will now plot a point on the curve by choosing an x-value and calculating its corresponding y-value. Let's try x=0.

When x=0, we have y=-8. As such, the point (0,-8) lies on the curve. Let's plot this point and reflect it across the axis of symmetry.

Notice that both points have the same y-coordinate.

Step 4

Finally, we will sketch the parabola which passes through the three points. Remember not to use a straightedge for this!