Precalculus : Graphing Quadratic Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Graphing Quadratic Functions

What is the y-intercept of the following equation?

Possible Answers:

Correct answer:

Explanation:

The y-intercept can by found by solving the equation when x=0. Thus,

Example Question #2 : Graphing Quadratic Functions

Determine the y intercept of , where  .

Possible Answers:

Correct answer:

Explanation:

In order to determine the y-intercept of , set 

Solving for y, when x is equal to zero provides you with the y coordinate for the intercept. Thus the y-intercept is .

Example Question #3 : Graphing Quadratic Functions

What is the -intercept of the function, 

?

Possible Answers:

Correct answer:

Explanation:

To find the -intercept we need to find the cooresponding  value when

Substituting  into our function we get the following:

Therefore, our -intercept is .

Example Question #4 : Graphing Quadratic Functions

What is the value of the -intercept of ?

Possible Answers:

The graph does not have a -intercept

Correct answer:

Explanation:

To find the -intercept we need to find the cooresponding  value when . Therefore, we substitute in  and solve:

Example Question #5 : Graphing Quadratic Functions

Find the vertex, roots, and the  value that the line of symmetry falls on of the function .

Possible Answers:

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on .

vertex , the roots  and , and the axis of symmetry would fall on x=.5.

vertex , the roots  and , and the axis of symmetry would fall on .

Correct answer:

vertex , the roots  and , and the axis of symmetry would fall on .

Explanation:

All quadratic functions have a vertex and many cross the x axis at points called zeros or roots. If we know the vertex and its zeros, quadratic functions become very easy to draw since the vertex is also a line of symmetry (the zeros are equidistant from the vertex on either side).

Factor the equation  to get  and . Thus, the roots are 3 and -2.

The vertex can be found by using .

simplify

.

 

The axis of symmetry is halfway between the two roots, or simply the x coordinate of the vertex. So the axis of symmetry lies on x=1/2. To graph, draw a point at the coordinate pair of the vertex. Then draw points on the x axis at the roots, and finally, trace upwards from the vertex through the roots with a gentle curve.

Example Question #6 : Graphing Quadratic Functions

Which of the following functions matches the provided parabolic graph?

 

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Possible Answers:

Correct answer:

Explanation:

Finding the vertex, intercept and axis of symmetry are crucial to finding the function that corresponds to the graph:

The vertex form of a quadratic function is written as:

and the coordinates for the vertex are:

Looking at the graph and the position of the axis of symmetry, the vertex is positioned at , leaving us with an equation so far of:

While we don't know a right away,  is the only option that really works. The y-intercept is at  and we can plug that into the formula to confirm that this is the correct function:

 

Example Question #7 : Graphing Quadratic Functions

Which of the following is an equation for the parabola represented in the graph below?

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Possible Answers:

Correct answer:

Explanation:

Immediately we can tell that the equation has a negative coefficient, because the parabola opens downward, forming an umbrella shape. Based upon the information given in the figure, we can use the intercepts, axis of symmetry, and the vertex to identify the equation of the parabola. Let's observe the vertex form of a parabola written as the following:

In this equation,  is the vertex of the parabola, and  determines whether the parabola opens upwards or downwards. The axis of symmetry is at  and the vertex is located at , which we can plug into the following function:

We know that  is negative because of the position of the parabola. 

 

 

Example Question #8 : Graphing Quadratic Functions

Which of the following is the given quadratic equation in vertex form?

Possible Answers:

Correct answer:

Explanation:

To solve for the vertex form, we must start by completing the square:

Example Question #1 : Express A Quadratic Function In Vertex Form

Which of the following is the appropriate vertex form of the following quadratic equation?

Possible Answers:

Correct answer:

Explanation:

This process outlines how to convert a quadratic function to vertex form:

Example Question #10 : Graphing Quadratic Functions

Give the coordinate pair of the vertex of this quadratic function .

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

Expressing quadratic functions in the vertex form is basically just changing the format of the equation to give us different information, namely the vertex. In order for us to change the function into this format we must have it in standard form . After that, our goal is to change the function into the form . We do so as follows:

subtract the constant over to the other side

halve the b term, square it, and add to both sides. 

Now factor the left side. 

now simplify the right side and move that number back over to the left side and you will be left with . I recommend looking up an example with numbers before you begin or at least recognizing that the fractions will end up being whole numbers in most problems. Below is specific explanation of the problem at hand. Try to use the generic equation to find the answer before following the step by step approach below.

 

move the constant over

halve the b term and add to both sides

factor the left side and simplify the right

move the constant over to achieve vertex form

 is the final answer with vertex at (-1,-7). Note that the formula is .

try this shortcut after you have mastered the steps: . Make sure you recognize that this formula gives you an x and y coordinate for the vertex and that each coordinate of the pair is fraction in the formula. This will give you the vertex of the equation if it is in standard form. However, don't rely on this as completing the square is also a method for finding the roots. So you need to know both methods before you cut the corner.

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