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The quadratic function is given by the following expression:
[tex]f(x)=-x^2-2x+3[/tex]The direction at which the graph opens is determined by the signal of the number multiplying x². If the number is positive then the graph opens upwards, if it is negative it opens downward. In this case it is negative so it opens donward.
The vertex of a quadratic expression can be found by the following expression:
[tex]x=\frac{-b}{2a}[/tex]Where a is the number multiplying "x²", while b is the number multiplying "x". Applying the data from the problem we have:
[tex]x=\frac{-(-2)}{2\cdot(-1)}=\frac{2}{-2}=-1[/tex]To find the value of "y" for the vertex we need to apply the coordinate for x on the expression. We have:
[tex]\begin{gathered} f(-1)=-(-1)^2-2\cdot(-1)+3 \\ f(-1)=-1+2+3=4 \end{gathered}[/tex]The coordinates of the vertex are (-1,4).
To sketch a graph we need to find the x-intercept and y-intercept of the function. These are given when f(x) = 0 and x=0 respectively. Let's find these points.
[tex]\begin{gathered} 0=-x^2-2x+3 \\ -x^2-2x+3=0 \\ x_{1,2}=\frac{-(-2)\pm\sqrt[]{(-2)^2-4(-1)(3)}}{2\cdot1} \\ x_1=-3 \\ x_2=1 \end{gathered}[/tex][tex]f(x)=-0^2-2\cdot0+3=3[/tex]The x intercept happens in two points -3 and 1, while the y intercept happens in the point 3. With this and the vertex we can sketch the function.
