Question

How do you use the important points to sketch the graph of `y=x^2+14x+29`?

Answer 1

Important points we need to sketch the graph of

`color(red)(y = x^2+14x+29)`

are

`color(blue)(Vertex = (-7, -20)`

`x`-Intercepts are `x = [-7+-2sqrt(5)]`

`y`-Intercept is at `(0,29)`

Explanation:

Given:

`color(red)(y = x^2+14x+29)`

Factoring Method is unsuitable for this problem situation.

To find the Solutions, we use the Quadratic Formula:

`color(green)((-b+- sqrt(b^2 - 4*a*c))/(2*a))`

We can see that

`color(blue)(a = 1; b = 14 and c= 29)`

`x = [-14+-sqrt(14^2-4*1*29)]/(2*1)`

`x = [-14+-sqrt(80)]/(2)`

`x = [-14+-sqrt(16*5)]/(2)`

`x = [-14+-4*sqrt(5)]/(2)`

`x = [-7+-2sqrt(5)]`

Therefore,

`x = [-7+-4.472136]`

`x = [-7+4.472136], [-7-4.472136]`

`x =-2.52786, -11.4721`

Hence,

x-Intercepts are: `-2.52786, -11.4721`

To get the y-intercept we substitute `x=0` in

`color(red)(y = x^2+14x+29)`

`:.` our y-intercept is at `(0, 29)`

Next, we will work on our Vertex

We can use the formula `color(blue)[[-b/(2*a)]` to find our Vertex

`rArr -14/(2.1)`

`rArr -7`

This is the x-coordinate value of our vertex

To find the y-coordinate value of our vertex, we substitute this value of `x` in `color(red)(y = x^2+14x+29)`

We get,

`y = (-7)^2 + 14*(-7) + 29`

Therefore,

`y = 49 -98 + 29`

`y = -20`

This is our y-coordinate value of our vertex.

Hence, our vertex is `color(green)[(-7, -20)`

Now, we have all the important points with us for the graph.

Analyze the graph below to understand better:

Hope you find the solution useful.