find the vertex find the axis of symmetry and find the x and y intercepts then draw a graph

Question

for the parabola y=-x^2-4x+5 find the vertex find the axis of symmetry and find the x and y intercepts then draw a graph

Answer 1

y = -x² - 4x + 5

 

We must write the equation in vertex form.

 

y = a(x - h)² + k

 

and the vertex has coordinates (h, k).

 

 

Since the leading term has a coefficient of -1, the value of a will be -1.

 

y = -1(x - h)²  + k

 

 

Now we determine the value of h.  We examine the original equation to write an equivalent.  (x - h) is a perfect square.

 

y = -1(x² + 4x + 4) + 9

 

x² + 4x + 4 is  a perfect square.

 

Let's check this guess by expanding.

 

y = -x² - 4x - 4 + 9

y = -x² - 4x + 5

 

We get back the original equation.  This checks out.  Therefore, the vertex form is

 

y = -1(x + 2)² + 9

 

This means that

 

h = -2

k = 9

 

The vertex is (-2, 9).

 

 

The axis of symmetry is

x = -2

 

 

We can use the vertex form to find the x-intercept and y-intercept.

 

To find the y-intercept, set x = 0.

 

y = -1(x + 2)² + 9

y = -1(0 + 2)² + 9

y = -1(2)² + 9

y = -4 + 9

y = 5

 

The y-intercept is (0, 5).

 

 

To find the x-intercept, set y = 0. 

 

y = -1(x + 2)² + 9

0 = -1(x + 2)² + 9

-9 = -1(x + 2)²

9 = (x + 2)²

 

±√(9) = x + 2

 

-2 ± √(9) = x

 

-2 + 3 = x            and          -2 - 3 = x

       1 = x             and                -5 = x

 

 

The x-intercepts are (-5, 0) and (1, 0).