| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
Here are a few recommended readings before getting started with this lesson.
Besides the intercept and the vertex forms, another essential and common form of a quadratic function is its standard form.
The standard form of a quadratic function is a quadratic function written in a specific format.
f(x)=ax2+bx+c
Here, a, b, and c are real numbers with a=0. The term with the highest degree — the quadratic term — is written first, then the linear term, followed by the constant term. The standard form of the function can be used to determine the direction of the parabola, the y-intercept, the axis of symmetry, and the vertex.
Direction of the Graph | Opens upward when a>0 |
---|---|
Opens downward when a<0 | |
y-intercept | c |
Axis of Symmetry | x=-2ab |
Vertex | (-2ab,f(-2ab)) |
Determine whether the given quadratic function is expressed in standard form.
a=1, b=-4
Identity Property of Multiplication
-b-a=ba
Calculate quotient
The y-intercept can be determined by using the constant term c of the given function. In this case, c is equal to 3. This means that the y-intercept occurs at (0,3).
The axis of symmetry divides the graph into two mirror images. Therefore, the reflection of the y-intercept across the axis of symmetry is also on the parabola.
If the vertex of a quadratic function lies on the y-axis, then any point that lies on the graph other than the vertex should be found and reflected across the axis of symmetry. In this case, the y-axis is the axis of symmetry.
Finally, connect the points with a smooth curve to graph the parabola.
The graph of the function opens upward. This is expected, since the value of a is 1, which is a positive number.
Find the coordinates of the vertex of the parabola that corresponds to the quadratic function written in standard form. If necessary, round the answer to 2 decimal places.
Zain lives in a quiet rural town where it seems not a lot happens. He decides to adopt a German shepherd puppy. After a few short weeks the puppy is growing so fast, they are afraid that soon the pup will be able to jump over the fence around their garden!
These steps will be done one at a time.
x=9
Calculate power and product
(-a)b=-ab
Add terms
The y-intercept is given by the constant term c of the function rule. In this case this term is equal to 0, which means that the y-intercept occurs at (0,0) — the origin.
Now, another point that lies on the parabola can be found by reflecting the y-intercept in the axis of symmetry.
The third point that lies on the parabola is at (18,0).
Finally, the points will be connected with a smooth curve to graph the parabolic shape. Since the function represents a dog's jump, negative values of the function will not be included.
Depending on the conditions, it is convenient to rewrite a quadratic function given in intercept or vertex form in its standard form.
Both the vertex and intercept forms of a quadratic function can always be rewritten in standard form.
Form | Equation | How to Rewrite? |
---|---|---|
Vertex Form | y=a(x−h)2+k | Expand (x−h)2, distribute a, and combine like terms. |
Intercept Form (also called Factored Form) |
y=a(x−p)(x−q) | Multiply a(x−p)(x−q) and combine like terms. |
The mayor of Zain's sleepy town wants to do something exciting for the community. Lucky her, she noticed an increasing interest in people taking up skateboarding, so she knew that building a new skate ramp would be a huge hit.
She was right! Now a local skateboard company named SuTeKi Sk8 sees this as an opportunity to make some big sales. SuTeKi Sk8 investigates the profit of its two coolest models, Sketchtastic and Sugoi Sketch. They collect data and find a relationship between the price and the profit on sales of both skateboards. The following quadratic functions represent their results.
Skateboard | Relationship |
---|---|
Sketchtastic | y=-0.015(x−50)2+37 |
Sugoi Sketch | y=-0.008(x−65)2+28 |
In these function rules, x is the price in dollars and y is the corresponding profit in thousands of dollars.
Skateboard | Vertex Form | Standard Form |
---|---|---|
Sketchtastic | y=-0.015(x−50)2+37 | y=-0.015x2+1.5x−0.5 |
Sugoi Sketch | y=-0.008(x−65)2+28 | y=-0.008x2+1.04x−5.8 |
Graphs:
Skateboard | Profit on Sales |
---|---|
Sketchtastic | ≈$23000 |
Sugoi Sketch | ≈$26000 |
Comparison: See solution.
Vertex Form: y=a(x−p)2+q | |
---|---|
Sketchtastic | Sugoi Sketch |
y=-0.015(x−50)2+37 | y=-0.008(x−65)2+28 |
(a−b)2=a2−2ab+b2
Calculate power and product
Distribute -0.015
Add terms
Sketchtastic | Sugoi Sketch | |
---|---|---|
Given Function | y=-0.015(x−50)2+37 | y=-0.008(x−65)2+28 |
Standard Form | y=-0.015x2+1.5x−0.5 | y=-0.008x2+1.04x−5.8 |
The functions can now be graphed using the standard form. To do so, there are five steps to follow.
The axis of symmetry is given by the equation x=2a-b, where a is the coefficient of the x2-term and b is the linear coefficient. Use a calculator to evaluate the equations if necessary.
Axis of Symmetry: x=2a-b | |
---|---|
y=-0.015x2+1.5x−0.5 | y=-0.008x2+1.04x−5.8 |
x=2(-0.015)-1.5 | x=2(-0.008)-1.04 |
x=50 | x=65 |
The axes of symmetry of the parabolas are the vertical lines with equations x=50 and x=65.
The vertices of the parabolas lie on the corresponding axes of symmetry. This means that x=50 and x=65 are the x-coordinates of the vertices. Now, these values will be substituted for x into the function rules to find the y-coordinates.
y=-0.015x2+1.5x−0.5 | y=-0.008x2+1.04x−5.8 | |
---|---|---|
x-coordinate | 50 | 65 |
y-coordinate | y=-0.015(50)2+1.5(50)−0.5 ⇕ y=37 |
y=-0.008(65)2+1.04(65)−5.8 ⇕ y=28 |
Vertex | (50,37) | (65,28) |
The vertices can now be added to the graph.
The y-intercepts are given by the constant term of the function rules.
Function Rule | y-intercept |
---|---|
y=-0.015x2+1.5x−0.5 | -0.5 |
y=-0.008x2+1.04x−5.8 | -5.8 |
Next, another point that lies on each parabola can be found by reflecting the y-intercept in the axis of symmetry.
Finally, the corresponding points will be connected with a smooth curve to graph the parabolic shapes.
The x-coordinate of each vertex is the price of the corresponding skateboard at which the profit of the company is maximized. This means that the vertex represents the optimal price and its corresponding profit. The company should use these values to make the highest return on sales.
From the graph, it can be seen that the profits of skateboards Sketchtastic and Sugoi Sketch are about 23 and 26 dollars in the thousands, respectively. To verify this, x=80 will be substituted in both function rules.
Sketchtastic | Sugoi Sketch | |
---|---|---|
x | 80 | 80 |
Substitute | -0.015(80)2+1.5(80)−0.5 | -0.008(80)2+1.04(80)−5.8 |
Evaluate | 23.5 | 26.2 |
The estimated values are slightly less than the exact values.
Graph:
x=65, y=487
Subtract terms
a(-b)=-a⋅b
acb=ca⋅c+b
LHS/(-975)=RHS/(-975)
Put minus sign in front of fraction
ba/c=b⋅ca
Use a calculator
Rearrange equation