Graphing Quadratic Functions
1️⃣ Understanding the Graph of a Quadratic Function
A quadratic function is any function that can be written in the form:
\[ f(x)=ax^2+bx+c \]
- ✔ The graph of a quadratic function is a parabola.
- ✔ If \( a>0 \), the parabola opens upwards (U-shape).
- ✔ If \( a<0 \), the parabola opens downwards (∩-shape).
Example: Graph \( f(x)=x^2-4x+3 \). Since \( a=1>0 \), the parabola opens upward.
2️⃣ Key Features of a Parabola
1️⃣ Vertex – The Turning Point
- ✔ The vertex is the highest or lowest point.
- ✔ It is given by the formula: \[ x=\frac{-b}{2a} \]
- ✔ Substitute \( x \) back into the function to find \( y \).
Example: Find the vertex of \( f(x)=x^2-6x+5 \):
\( x=\frac{6}{2}=3 \)
Then, \( f(3)=9-18+5=-4 \). Vertex = \( (3,-4) \).
2️⃣ Axis of Symmetry
The vertical line through the vertex, given by the same formula \( x=\frac{-b}{2a} \), divides the parabola into mirror images.
Example: For \( f(x)=2x^2-4x+1 \), the vertex is \( (1,-1) \) so the axis of symmetry is \( x=1 \).
3️⃣ X-Intercepts (Roots)
The x-intercepts occur where \( f(x)=0 \). Solve the equation \( ax^2+bx+c=0 \) to find them.
Example: Find the x-intercepts of \( f(x)=x^2-5x+6 \):
\( (x-2)(x-3)=0 \)
Thus, \( x=2 \) and \( x=3 \). X-intercepts: \( (2,0) \) and \( (3,0) \).
4️⃣ Y-Intercept
The y-intercept is found by evaluating \( f(0) \).
Example: For \( f(x)=3x^2-5x+2 \), \( f(0)=2 \). Y-intercept: \( (0,2) \).
3️⃣ How to Graph a Quadratic Function
Example: Graph \( f(x)=x^2-4x+3 \):
- Find the vertex: \( x=\frac{-(-4)}{2(1)}=\frac{4}{2}=2 \) and \( f(2)=4-8+3=-1 \). Vertex = \( (2,-1) \).
- Find the x-intercepts: Factor to get \( (x-1)(x-3)=0 \); thus, \( x=1 \) and \( x=3 \). X-intercepts: \( (1,0) \) and \( (3,0) \).
- Find the y-intercept: \( f(0)=3 \). Y-intercept: \( (0,3) \).
- Plot the points and draw a smooth parabola.
Practice Questions – Graph and Analyze
- Find the vertex, axis of symmetry, x-intercepts, and y-intercept for: \[ f(x)=x^2-6x+8 \]
- Find the axis of symmetry and graph: \[ f(x)=2x^2-4x+1 \]
- Find the x-intercepts of: \[ f(x)=-x^2+7x-12 \]
- Determine the vertex and graph: \[ f(x)=x^2+4x+5 \]
- Find the y-intercept of: \[ f(x)=3x^2-5x+2 \]
- Identify if the function opens upward or downward and sketch: \[ f(x)=-2x^2+5x-3 \]
- Find the range of: \[ f(x)=-x^2+6x-9 \]
- Solve for \( x \) where \( f(x)=0 \) in: \[ f(x)=4x^2-16x+15 \]
- Determine the maximum or minimum value for: \[ f(x)=-3x^2+12x-4 \]
- Write an equation for a parabola with vertex at \( (3,2) \) and passing through \( (0,5) \).
Challenging Problems – Push Your Limits!
- The function \( h(t)=-5t^2+20t+30 \) models the height of a projectile. Find the maximum height.
- Find the equation of a quadratic function with x-intercepts \( x=2 \) and \( x=5 \) and a y-intercept of 10.
- A farmer is fencing a rectangular garden next to a river using 60 m of fencing (no fence is needed along the river). Find the maximum area he can enclose.
- If a ball is thrown upward with a velocity of 30 m/s, its height is \( h(t)=-4.9t^2+30t \). Find when the ball hits the ground.
- The profit function is \( P(x)=-2x^2+8x+15 \). Find the maximum profit and the number of items sold to achieve it.
7️⃣ Summary
- ✅ A quadratic function graphs as a parabola.
- ✅ Key features include the vertex, axis of symmetry, x-intercepts, and y-intercept.
- ✅ Graphing quadratics helps visualize maximum and minimum values.
8️⃣ Fantastic Job!
Great work! Ready to explore the next advanced topic? Let's move on to more applications of quadratic functions!