Chapter Review
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Identify the vertex first. Then use it to find the axis of symmetry.
Minimum Value: - 4
Decreasing Interval: To the left of x=1
Increasing Interval: To the right of x=1
Note that the function is already written in vertex form, f(x)=a(x-h)^2+k, where a, h, and k are either positive or negative numbers. f(x)=3(x-1)^2-4 To solve the problem and draw the graph, we will follow four steps.
We will first identify the constants a, h, and k. Recall that if a<0, the parabola will open downwards. Conversely, if a>0, the parabola will open upwards. Vertex Form:& f(x)=a(x-h)^2+k Function:& f(x)=3(x-1)^2+(- 4) We can see that a=3, h=1, and k=- 4. Since a is grater than 0, the parabola will open upwards.
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 (1,- 4). Therefore, the axis of symmetry is the vertical line x=1.
Since a=3 is greater than 0, the function decreases to the left of the minimum value and increases to the right of the minimum value, which we know occurs at x=1. This means that f(1)=- 4 will be the maximum value of f. Minimum value:& - 4 Decreasing Interval:& To the left of 1 Increasing Interval:& To the right of 1
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 f(0)=- 1. Thus, the point (0,- 1) lies on the curve. Let's plot this point and reflect it across the axis of symmetry.
Note that both points have the same y-coordinate. Finally, we will sketch the parabola which passes through the three points. Remember not to use a straightedge for this!
