AI Math Solver
Resources
Questions
Pricing
Login
Register
Home
>
Questions
>
Find Vertex and Opening Direction of Quadratic Function $f(x)=x^2+6x+15$
Mathematics
Grade 9 (Junior High School)
Question Content
For the quadratic function $f(x) = x^2 + 6x + 15$, find its vertex and determine its opening direction.
Correct Answer
Vertex: $(-3, 6)$; Opening Direction: Upwards
Detailed Solution Steps
1
Step 1: Recall the vertex form of a quadratic function: $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex, and $a$ determines the opening direction.
2
Step 2: Convert the given standard form $f(x)=x^2+6x+15$ to vertex form by completing the square. First, group the $x$-terms: $f(x)=(x^2+6x)+15$.
3
Step 3: Complete the square for the $x$-terms: take half of the coefficient of $x$ (which is $6$, half is $3$), square it ($3^2=9$). Add and subtract this value inside the parentheses: $f(x)=(x^2+6x+9-9)+15$.
4
Step 4: Rewrite the perfect square trinomial and simplify the constants: $f(x)=(x+3)^2 -9 +15=(x+3)^2+6$. Now it is in vertex form, so $h=-3$ and $k=6$, meaning the vertex is $(-3,6)$.
5
Step 5: Identify the coefficient $a$ from the original function: $a=1$. Since $a>0$, the parabola opens upwards.
Knowledge Points Involved
1
Quadratic Function Standard Form
The standard form of a quadratic function is $f(x)=ax^2+bx+c$, where $a$, $b$, $c$ are constants and $a\\neq0$. This form is used for basic identification of quadratic functions, and $a$ directly determines the opening direction of the parabola.
2
Vertex Form of Quadratic Function
The vertex form is $f(x)=a(x-h)^2+k$, where $(h,k)$ is the vertex of the parabola. This form allows direct reading of the vertex coordinates, which is useful for analyzing the minimum or maximum value of the quadratic function.
3
Completing the Square
Completing the square is an algebraic method used to convert a quadratic expression from standard form to vertex form. It involves creating a perfect square trinomial by adding and subtracting the square of half the coefficient of the linear $x$-term.
4
Opening Direction of Quadratic Function Parabola
For a quadratic function $f(x)=ax^2+bx+c$, if $a>0$, the parabola opens upwards (has a minimum value at the vertex); if $a<0$, the parabola opens downwards (has a maximum value at the vertex).
Loading solution...
