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Question

draw an acurate graph of the quadratic function y=x^2 -7x + 10 on the grid.fill in the table of values for five critical points (vertex,x-intercepts(s),y intercepts,and the symmetrical point with the y-intercept) then answer

 

a) explain how to use your graph to find the solution to the quadratic equation x^2 -7x + 10 = 0

 

b) show how to check the answers you obtained from the graph to determine whether you have the correct answers.

 

Thanks.

Answer 1

Note :

To graph a quadratic function follow the steps :

Step 1 : Find the equation of the axis of symmetry.

Step 2 : Find vertex, and determine weather it is a maximum or minimum.

Step 3 : Find the y and x - intercepts.

Step 4 : Use symmetry to find additional points on the graph, if necessary.

Step 5 : Connect the points with a smooth curve.

The function is f(x) = y = x² - 7x + 10.

The standard form of quadratic function is f(x) = ax² + bx + c.

Step 1 :

Find the axis of symmetry :

Formula for the equation of the axis of symmetry:  x = - b/2a.

Substitute the values of b = - 7 and a = 1 in the formula, x = - b/2a.

x = - (- 7)/2(1) = 7/2.

The equation for the axis of symmetry is x = 7/2.

Step 2 :

Find the vertex :

To find the vertex, use the value of equation for the axis of symmetry as the  x - coordinate of the vertex.

To find the y - coordinate, substitute the value of x = 7/2 in the original function, y = x² - 7x + 10.

y = (7/2)² - 7(7/2) + 10

y = 49/4 - 49/2 + 10

y = - 9/4.

The vertex is (7/2, - 9/4).

Determine whether the function has maximum or minimum value :

The value of a = 1 > 0 (positive), so the graph of function opens upward and has a minimum value.

The minimum value (y - coordinate of the vertex) is - 9/4.

Step 3 :

Find the y -  intercept :

To find the y - intercept, Substitute the value x = 0 in the original function, f(x) = y = x² - 7x + 10.

y = (0)² - 7(0) + 10

y = 10.

The y - intercept is 10.

The point is (0, 10).

Find the x -  intercept :

To find the x - intercept, Substitute the value y = 0 in the original function, f(x) = y = x² - 7x + 10.

x² - 7x + 10 = 0

x² - 5x - 2x + 10 = 0

x(x - 5) - 2(x - 5) = 0

(x - 2)(x - 5) = 0

The x - intercept is 2 and 5.

The points are (2, 0) and (5, 0).

Step 4 :

• Since, The axis of symmetry divides the parabola into two equal parts.So, if there is a point (0, 10) on one side, there is a corresponding point on other side that is the same distance from the axis of symmetry and has the same y - value.
• The distance between the points (0, 10) and (7/2, 10) = 3.5 = The distance between (7/2, - 9/4) and the point (x, 10) paired with it on other side of the axis of symmetry and has the same y - value.
• The distance between (7/2, 10) and the point paired with it on other side of the axis of symmetry. = 7/2 + 3.5 = 7.
• Therefore, The point paired with it on other side of the axis of symmetry is (7, 10).
• Connect these points and create a smooth curve.

Graph :