Given: A quadratic equation
[tex]f(x)=x^2-2x-15[/tex]Required: To find the vertex, x-intercepts, y-intercept, and graph the given quadratic function.
Explanation: Comparing the given equation with general quadratic function
[tex]f(x)=ax^2+bx+c\text{ }[/tex]we get, a=1, b=-2, and c=-15. Now the x coordinate of the vertex of the quadratic function is
[tex]x=-\frac{b}{2a}[/tex]Hence,
[tex]x=1[/tex]At x=1, f(x) is
[tex]\begin{gathered} f(1)=(1)^2-2(1)-15 \\ =-16 \end{gathered}[/tex]Hence the vertex of the given function is (1,-16). Now for getting the x-intercept we put f(x)=0, i.e.,
[tex]\begin{gathered} x^2-2x-15=0 \\ (x-5)(x+3)=0 \\ x=5\text{ and} \\ x=-3 \end{gathered}[/tex]Hence the x-intercepts are (5,0) and (-3,0). Similarly, for y-intercept, we put x=0 and find f(x) as follows
[tex]\begin{gathered} f(0)=0^2-2(0)-15 \\ =-15 \end{gathered}[/tex]Hence y-intercept is (0,-15). Now using the vertex and intercepts to graph the given quadratic function is shown below.
Final Answer: b) Vertex=(1,-16)
c) x intercepts are (5,0) and (-3,0)
d) y intercepts is (0,-15)
e) The graph of f(x) is shown below.
