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The Vertex Form of a quadratic function is :
color(blue)(y=f(x)=a(x-h)^2+k, where color(green)(( h, k ) is the Vertex of the Parabola.
Quadratic Function is given in Vertex Form: color(red)(y = (x + 5)^2 - 3
color(brown)(h=-5 and k = -3
Vertex is at color(green)((h,k): color(blue)((-5, -3)
![enter image source here]()
Plot the Vertex and the color(red)((x,y) values from the data table.
To find the x-intercepts:
f(x)=(x+5)^2-3
Let f(x)=0
:. (x+5)^2-3 = 0
Add color(red)(3 to both sides:
(x+5)^2-3+color(red)(3)= 0+color(red)(3
(x+5)^2-cancel 3+color(red)(cancel 3)= 0+color(red)(3
(x+5)^2 = 3
Take Square root on both sides to simplify:
sqrt((x+5)^2) = sqrt(3)
(x+5) = +- sqrt(3)
Subtract color(red)(5) from both sides:
(x+5)-color(red)(5) = +- sqrt(3)-color(red)(5)
(x+cancel 5)-color(red)(cancel 5) = +- sqrt(3)-color(red)(5)
x=+-sqrt(3)-5
Hence, color(blue)(x=[sqrt(3)-5] is one solution and color(blue)(x=[-sqrt(3)-5] is the other.
Using a calculator,
color(blue)(x~~ -3.26795) is one solution.
color(blue)(x~~ -6.73205) is another solution.
Hence, x-intercepts are: x~~ -3.3, x~~ -6.7
Verify this solution by using graphs:
color(green)("Graph 1"
Graph of color(blue)(y=x^2
This is the Parent Graph.
![enter image source here]()
Use this graph to understand the behavior of the given quadratic function.
color(green)("Graph 2"
Graph of color(blue)(y = (x + 5)^2 - 3
Study the graphs of both the Parent function and the given function.
![enter image source here]()
Next, verify the x-intercepts:
![enter image source here]()
Hope it helps.
