Answer :
Answer:
See the below works.
Step-by-step explanation:
We can graph the function f(x) = -16x² + 24x + 16 by these steps:
- find the x-intercepts, where the value of f(x) = 0
- find the y-intercept, where the value of x = 0
- determine the direction of the opening and find the vertex
- (optional) find 2 additional coordinates that are reasonably far from the left and right of the above points to make a better graph
Part A:
When the graph intersects the x-axis, the y-value (f(x)) equals to 0. Therefore, to find the x-intercepts, we substitute f(x) with 0:
[tex]\begin{aligned}f(x)&=-16x^2+24x+16\\0&=-16x^2+24x+16\\0&=2x^2-3x-2\\0&=(2x+1)(x-2)\\x&=-\frac{1}{2} \ or\ 2\end{aligned}[/tex]
[tex]\texttt{Hence, the coordinates }=(-\frac{1}{2} ,0)\ \texttt{and}\ (2,0)[/tex]
Part B:
To determine the opening direction of the graph (quadratic function), we apply these rules:
- If the coefficient of x² < 0 ⇒ the graph opens downwards and the vertex is a maximum
- If the coefficient of x² > 0 ⇒ the graph opens upwards and the vertex is a minimum
For this question, the coefficient of x² is -16, which is smaller than 0, then the graph opens downwards and the vertex is a maximum.
To find the vertex, we use this formula:
[tex]\boxed{(x,y)=\left(-\frac{b}{2a} ,\frac{-(b^2-4ac)}{4a} \right)}[/tex]
[tex]\begin{aligned}(x,y)&=\left(-\frac{24}{2(-16)} ,\frac{-(24^2-4(-16)(16))}{4(-16)} \right)\\\\&=\left(\frac{3}{4} ,25\right)\end{aligned}[/tex]
Part C:
Refer to the top part for the steps to graph f(x). From Part A and B, we have the x-intercepts and the vertex. Now, we need to find the y-intercept. When the graph intersect the y-axis, the x-value equals to 0. Therefore, to find the y-intercepts, we substitute x with 0:
[tex]\begin{aligned}f(x)&=-16x^2+24x+16\\&=-16(0)^2+24(0)+16\\&=16\end{aligned}[/tex]
[tex]\texttt{Hence, the coordinates }=(0,16)[/tex]
Now, we have all 4 coordinates needed and we can connect them to create the graph. (see the attached drawing)
Note:
To make a nicer graph, we can find 2 coordinates that are reasonably far from the known coordinates. Since the most left point is [tex](-\frac{1}{2} ,0)[/tex] and the most right point is [tex](2,0)[/tex], then we can find coordinates with x = -1 and 3 to make a nicer graph.
[tex]f(-1)=-16(-1)^2+24(-1)+16=-24[/tex]
[tex]f(3)=-16(3)^2+24(3)+16=-56[/tex]
Then by connecting with points (-1, -24) and (3, -56) will make the graph looks nicer.
![View image karmenchong](https://us-static.z-dn.net/files/d54/87d5a8365ba3d48448ad6a7f932cf92f.png)