B2. The quadratic function $h$ has its vertex at the intersection of the graphs of the functions $f(x)=\log _{3} x$ and $g(x)=-x+4$ and has an initial value of -17. Write the formula of the quadratic function $h$ in standard and vertex form. Draw the graphs of both functions $f$ and $g$ in the same Cartesian coordinate system.

Naloge za 3. letnik
B2. Using the graphs of the functions $f(x)=\log _{3} x$ and $g(x)=-x+4$, we find the vertex of the desired quadratic function, which is $T(3,1)$. We substitute the coordinates of the vertex and initial values into the vertex form $h(x)=a(x-p)^{2}+q$ and calculate $a=-2$. We write the vertex form as $h(x)=-2(x-3)^{2}+1$ and

the standard form as $h(x)=-2 x^{2}+12 x-17$.

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=69&width=1753&top_left_y=1022&top_left_x=160)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=62&width=1753&top_left_y=1074&top_left_x=160)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=63&width=1756&top_left_y=1122&top_left_x=161)

Writing or using the vertex form $h(x)=a(x-p)^{2}+q \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=60&width=1753&top_left_y=1226&top_left_x=160)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-23.jpg?height=52&width=1756&top_left_y=1276&top_left_x=161)

Writing the vertex form $h(x)=-2(x-3)^{2}+1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

Writing the standard form $h(x)=-2 x^{2}+12 x-17 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$