Math Problem Statement

h(x)=x2+9x+18 Step 1 of 3: Find the vertex.

Solution

To find the vertex of the quadratic function h(x)=x2+9x+18h(x) = x^2 + 9x + 18, we can use the vertex formula for a parabola in the form ax2+bx+cax^2 + bx + c. The x-coordinate of the vertex is given by:

xvertex=b2ax_{\text{vertex}} = -\frac{b}{2a}

For the given function h(x)=x2+9x+18h(x) = x^2 + 9x + 18, we identify:

  • a=1a = 1
  • b=9b = 9

Now, substitute these values into the formula:

xvertex=92(1)=92=4.5x_{\text{vertex}} = -\frac{9}{2(1)} = -\frac{9}{2} = -4.5

Next, to find the y-coordinate of the vertex, substitute xvertex=4.5x_{\text{vertex}} = -4.5 back into the original equation h(x)h(x):

h(4.5)=(4.5)2+9(4.5)+18h(-4.5) = (-4.5)^2 + 9(-4.5) + 18 h(4.5)=20.2540.5+18=2.25h(-4.5) = 20.25 - 40.5 + 18 = -2.25

Thus, the vertex of the function is (4.5,2.25)(-4.5, -2.25).

Would you like more details or have any questions about this?

Here are some related questions:

  1. How do you find the axis of symmetry of a quadratic function?
  2. What does the vertex tell us about the graph of the parabola?
  3. How can you determine if the vertex is a maximum or a minimum?
  4. Can you rewrite this quadratic function in vertex form?
  5. How do the coefficients of a quadratic function affect its graph?

Tip: The vertex of a parabola is its turning point, which can either be the maximum or minimum depending on the direction the parabola opens.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Vertex of a Parabola

Formulas

x_vertex = -b/2a
h(x) = ax^2 + bx + c

Theorems

Vertex formula for quadratic functions

Suitable Grade Level

Grades 9-10