Vertex form is an algebraic format used to express quadratic function rules.
y=a(x−h)2+k
In this form, a gives the direction of the parabola. When a>0, the parabola faces upward and when a<0, it faces downward. The vertex of the parabola lies at (h,k), and the axis of symmetry is x=h. Consider the graph of f(x)=-21(x+4)2+8.
From the graph, we can connect the following characterisitcs to the function rule. directionvertexaxis of symmetry:downward:(-4,8):x=-4→a<0→h=-4, k=8→h=-4
Notice that although the factor in the function rule shows (x+4)2, h is actually equal to -4. This coincides with a horizontal translation of a quadratic function.Writing quadratic functions in vertex form, y=a(x−h)2+k, is advantageous because it clearly presents three characteristics of the function. directionvertexaxis of symmetry: a>0⇒upward, a<0⇒downward:(h,k):x=h It is possible to graph a quadratic function using its characteristics. Consider the function y=(x−2)2−4.
To begin, identify the vertex, (h,k), from the function rule. Since the rule is y=(x−2)2−4, h=2andk=-4. Thus, the vertex is (2,-4). Next, plot the vertex on a coordinate plane.
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror images. Since the axis of symmetry is x=h, here it is x=2.
The axis of symmetry divides the parabola into two mirror images. Thus, points on one side of the parabola can be reflected across the axis of symmetry. Notice that the y-intercept is 2 units away from the axis of symmetry.
There exists another point directly across from the y-intercept that is also 2 units from the axis of symmetry.
Now, with three points plotted, the general shape of the parabola can be seen. It appears that the parabola faces upward. Since a=1 in the given function rule, this should be expected. To draw the parabola, connect the points with a smooth curve.
The characteristics of two functions, f(x) and g(x), are described below. Determine which of the function rules could represent g and h. -Axis of symmetryVertexf(x).x=1(1,3)g(x)-x=-1(-1,-3)
Number | Function rule |
---|---|
1 | y=-(x−1)2+3 |
2 | y=(x−1)2−3 |
3 | y=-(x−1)2−3 |
4 | y=(x−1)2+3 |
5 | y=(x+1)2+3 |
6 | y=(x+1)2−3 |
7 | y=-(x+1)2+3 |
8 | y=-(x+1)2−3 |
When a quadratic function is given in vertex form, y=a(x−h)2+k, both the vertex and the axis of symmetry can be easily seen. vertexaxis of symmetry:(h,k):x=h It is given that the axis of symmetry for g(x) is x=1 and for f(x) it is x=-1. Since the axis of symmetry is given as x=h, this gives h=1 for g and h=-1 for f. Thus, we can begin to write the function rules for g(x) and f(x) in vertex form as follows. Note that (x−(-1)) becomes (x+1). g(x)=a(x+1)2+kf(x)=a(x−1)2+k We can use the given vertices to determine the values of k. The vertex of g is (-1,-3). This gives k=-3. Similarly, f(x)'s vertex (1,3) gives k=3. Thus, we can write the rules as f(x)g(x)=a(x+1)2+3=a(x−1)2−3. Comparing these rules to the table above, it can be seen that rule number 2 and 3 can represent g(x) and 5 and 7 can represent f(x).
Number | Function rule |
---|---|
2 | y=(x−1)2−3 |
3 | y=-(x−1)2−3 |
5 | y=(x+1)2+3 |
7 | y=-(x+1)2+3 |
There are two candidates remaining for each function. The rules with a=-1 correspond to a downward parabola, whereas the rules with a=1 correspond to an upward parabola. To determine if any option is completely correct, we'd need more information about the parabola, specifically its direction or if its vertex is a minimum or a maximum.
Write the rule for the quadratic function that has the vertex (5,-1) and passes through the point (10,-11).
Since the parabola passes through the given points, the created rule is correct.