Math Problem Statement

Graph the quadratic function y = x^2 + 6x + 7.

Solution

The function given in the image is:

y=x2+6x+7y = x^2 + 6x + 7

This is a quadratic function, and its graph will be a parabola. Let's break down how to graph it:

Steps to Graph:

  1. Identify the key features:

    • Vertex: The vertex form of a quadratic function is y=a(xh)2+ky = a(x-h)^2 + k, where (h,k)(h, k) is the vertex. You can find the vertex using the formula x=b2ax = -\frac{b}{2a}. Here, a=1a = 1 and b=6b = 6, so: x=62(1)=3x = -\frac{6}{2(1)} = -3 Now, substitute x=3x = -3 into the equation to find the yy-coordinate of the vertex: y=(3)2+6(3)+7=918+7=2y = (-3)^2 + 6(-3) + 7 = 9 - 18 + 7 = -2 The vertex is at (3,2)(-3, -2).
  2. Y-intercept: The y-intercept occurs when x=0x = 0: y=02+6(0)+7=7y = 0^2 + 6(0) + 7 = 7 So, the y-intercept is at (0,7)(0, 7).

  3. Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, which is x=3x = -3.

  4. Additional Points: To accurately plot the graph, choose other values of xx, such as x=2x = -2 and x=4x = -4, and compute the corresponding yy-values:

    • For x=2x = -2, y=(2)2+6(2)+7=412+7=1y = (-2)^2 + 6(-2) + 7 = 4 - 12 + 7 = -1.
    • For x=4x = -4, y=(4)2+6(4)+7=1624+7=1y = (-4)^2 + 6(-4) + 7 = 16 - 24 + 7 = -1.
  5. Plot the points:

    • Vertex: (3,2)(-3, -2)
    • Y-intercept: (0,7)(0, 7)
    • Points: (2,1)(-2, -1) and (4,1)(-4, -1)
  6. Draw the parabola: Since a=1a = 1 (positive), the parabola opens upwards. Use the points to sketch a smooth curve.

Would you like further details or help with graphing? Here are some related questions:

  1. How do you convert a quadratic equation to vertex form?
  2. How do you determine whether a parabola opens upwards or downwards?
  3. What is the significance of the axis of symmetry in a quadratic graph?
  4. How do you find the roots of a quadratic equation?
  5. How do changes in the coefficients of x2x^2, xx, and the constant affect the graph?

Tip: The vertex provides a quick way to know the minimum or maximum point of the parabola, depending on its orientation!

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Graphing Parabolas
Algebra

Formulas

Standard form of quadratic function: y = ax^2 + bx + c
Vertex formula: x = -b / (2a)
y-intercept at x = 0

Theorems

Parabolas and Symmetry
Vertex Theorem

Suitable Grade Level

Grades 9-10